Calculus Examples

$f (x) = \frac{x}{1 + x^{4}}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 1 + x^{4}$ .

$\frac{(1 + x^{4}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(1 + x^{4}) \cdot 1 - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 1.2.2

Multiply $1 + x^{4}$ by $1$ .

$\frac{1 + x^{4} - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 1.2.3

By the Sum Rule, the derivative of $1 + x^{4}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}]$ .

$\frac{1 + x^{4} - x (\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 1.2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1 + x^{4} - x (0 + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 1.2.5

Add $0$ and $\frac{d}{d x} [x^{4}]$ .

$\frac{1 + x^{4} - x \frac{d}{d x} [x^{4}]}{{(1 + x^{4})}^{2}}$

Step 1.2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1 + x^{4} - x (4 x^{3})}{{(1 + x^{4})}^{2}}$

Step 1.2.7

Multiply $4$ by $- 1$ .

$\frac{1 + x^{4} - 4 x \cdot x^{3}}{{(1 + x^{4})}^{2}}$

Step 1.3

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 1.3.1

Move $x^{3}$ .

$\frac{1 + x^{4} - 4 (x^{3} x)}{{(1 + x^{4})}^{2}}$

Step 1.3.2

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 1.3.2.1

Raise $x$ to the power of $1$ .

$\frac{1 + x^{4} - 4 (x^{3} x^{1})}{{(1 + x^{4})}^{2}}$

Step 1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 + x^{4} - 4 x^{3 + 1}}{{(1 + x^{4})}^{2}}$

Step 1.3.3

Add $3$ and $1$ .

$\frac{1 + x^{4} - 4 x^{4}}{{(1 + x^{4})}^{2}}$

Step 1.4

Subtract $4 x^{4}$ from $x^{4}$ .

$\frac{1 - 3 x^{4}}{{(1 + x^{4})}^{2}}$

Step 1.5

Reorder terms.

$\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

$f'' (x) = \frac{{(x^{4} + 1)}^{2} \frac{d}{d x} (- 3 x^{4} + 1) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{({(x^{4} + 1)}^{2})}^{2}}$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

Multiply the exponents in ${({(x^{4} + 1)}^{2})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} \frac{d}{d x} (- 3 x^{4} + 1) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{2 \cdot 2}}$

Step 2.2.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} \frac{d}{d x} (- 3 x^{4} + 1) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.2.2

By the Sum Rule, the derivative of $- 3 x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (\frac{d}{d x} (- 3 x^{4}) + \frac{d}{d x} (1)) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 3 \frac{d}{d x} x^{4} + \frac{d}{d x} (1)) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 3 (4 x^{3}) + \frac{d}{d x} (1)) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.2.5

Multiply $4$ by $- 3$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3} + \frac{d}{d x} (1)) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3} + 0) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.2.7

Add $- 12 x^{3}$ and $0$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) \frac{d}{d x} {(x^{4} + 1)}^{2}}{{(x^{4} + 1)}^{4}}$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{4} + 1$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{4} + 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) (2 u \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 2.3.3

Replace all occurrences of $u$ with $x^{4} + 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - (- 3 x^{4} + 1) (2 (x^{4} + 1) \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 2.4

Differentiate.

Tap for more steps...

Step 2.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) \frac{d}{d x} (x^{4} + 1))}{{(x^{4} + 1)}^{4}}$

Step 2.4.2

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (1)))}{{(x^{4} + 1)}^{4}}$

Step 2.4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (4 x^{3} + \frac{d}{d x} (1)))}{{(x^{4} + 1)}^{4}}$

Step 2.4.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (4 x^{3} + 0))}{{(x^{4} + 1)}^{4}}$

Step 2.4.5

Simplify the expression.

Tap for more steps...

Step 2.4.5.1

Add $4 x^{3}$ and $0$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) ((x^{4} + 1) (4 x^{3}))}{{(x^{4} + 1)}^{4}}$

Step 2.4.5.2

Move $4$ to the left of $x^{4} + 1$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 2 (- 3 x^{4} + 1) (4 \cdot ((x^{4} + 1) x^{3}))}{{(x^{4} + 1)}^{4}}$

Step 2.4.5.3

Multiply $4$ by $- 2$ .

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) - 8 (- 3 x^{4} + 1) ((x^{4} + 1) x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) + (- 8 (- 3 x^{4}) - 8 \cdot 1) ((x^{4} + 1) x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.2

Apply the distributive property.

$f'' (x) = \frac{{(x^{4} + 1)}^{2} (- 12 x^{3}) + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3

Simplify the numerator.

Tap for more steps...

Step 2.5.3.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 12 {(x^{4} + 1)}^{2} x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.2

Rewrite ${(x^{4} + 1)}^{2}$ as $(x^{4} + 1) (x^{4} + 1)$ .

$f'' (x) = \frac{- 12 ((x^{4} + 1) (x^{4} + 1)) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.3

Expand $(x^{4} + 1) (x^{4} + 1)$ using the FOIL Method.

Tap for more steps...

Step 2.5.3.1.3.1

Apply the distributive property.

$f'' (x) = \frac{- 12 (x^{4} (x^{4} + 1) + 1 (x^{4} + 1)) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.3.2

Apply the distributive property.

$f'' (x) = \frac{- 12 (x^{4} x^{4} + x^{4} \cdot 1 + 1 (x^{4} + 1)) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.3.3

Apply the distributive property.

$f'' (x) = \frac{- 12 (x^{4} x^{4} + x^{4} \cdot 1 + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.4

Simplify and combine like terms.

Tap for more steps...

Step 2.5.3.1.4.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.4.1.1

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.4.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 12 (x^{4 + 4} + x^{4} \cdot 1 + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.4.1.1.2

Add $4$ and $4$ .

$f'' (x) = \frac{- 12 (x^{8} + x^{4} \cdot 1 + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.4.1.2

Multiply $x^{4}$ by $1$ .

$f'' (x) = \frac{- 12 (x^{8} + x^{4} + 1 x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.4.1.3

Multiply $x^{4}$ by $1$ .

$f'' (x) = \frac{- 12 (x^{8} + x^{4} + x^{4} + 1 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.4.1.4

Multiply $1$ by $1$ .

$f'' (x) = \frac{- 12 (x^{8} + x^{4} + x^{4} + 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.4.2

Add $x^{4}$ and $x^{4}$ .

$f'' (x) = \frac{- 12 (x^{8} + 2 x^{4} + 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.5

Apply the distributive property.

$f'' (x) = \frac{(- 12 x^{8} - 12 (2 x^{4}) - 12 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.6

Simplify.

Tap for more steps...

Step 2.5.3.1.6.1

Multiply $2$ by $- 12$ .

$f'' (x) = \frac{(- 12 x^{8} - 24 x^{4} - 12 \cdot 1) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.6.2

Multiply $- 12$ by $1$ .

$f'' (x) = \frac{(- 12 x^{8} - 24 x^{4} - 12) x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.7

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{8} x^{3} - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.8

Simplify.

Tap for more steps...

Step 2.5.3.1.8.1

Multiply $x^{8}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.8.1.1

Move $x^{3}$ .

$f'' (x) = \frac{- 12 (x^{3} x^{8}) - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.8.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 12 x^{3 + 8} - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.8.1.3

Add $3$ and $8$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{4} x^{3} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.8.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.8.2.1

Move $x^{3}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 (x^{3} x^{4}) - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{3 + 4} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.8.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (- 8 (- 3 x^{4}) - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.9

Simplify each term.

Tap for more steps...

Step 2.5.3.1.9.1

Multiply $- 3$ by $- 8$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8 \cdot 1) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.9.2

Multiply $- 8$ by $1$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{4} x^{3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.10

Simplify each term.

Tap for more steps...

Step 2.5.3.1.10.1

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.10.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{4 + 3} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.10.1.2

Add $4$ and $3$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{7} + 1 x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.10.2

Multiply $x^{3}$ by $1$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + (24 x^{4} - 8) (x^{7} + x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.11

Expand $(24 x^{4} - 8) (x^{7} + x^{3})$ using the FOIL Method.

Tap for more steps...

Step 2.5.3.1.11.1

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{4} (x^{7} + x^{3}) - 8 (x^{7} + x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.11.2

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{4} x^{7} + 24 x^{4} x^{3} - 8 (x^{7} + x^{3})}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.11.3

Apply the distributive property.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{4} x^{7} + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12

Simplify and combine like terms.

Tap for more steps...

Step 2.5.3.1.12.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.12.1.1

Multiply $x^{4}$ by $x^{7}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.12.1.1.1

Move $x^{7}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 (x^{7} x^{4}) + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{7 + 4} + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12.1.1.3

Add $7$ and $4$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 x^{4} x^{3} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12.1.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.12.1.2.1

Move $x^{3}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 (x^{3} x^{4}) - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 x^{3 + 4} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12.1.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 24 x^{7} - 8 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.1.12.2

Subtract $8 x^{7}$ from $24 x^{7}$ .

$f'' (x) = \frac{- 12 x^{11} - 24 x^{7} - 12 x^{3} + 24 x^{11} + 16 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.2

Add $- 12 x^{11}$ and $24 x^{11}$ .

$f'' (x) = \frac{12 x^{11} - 24 x^{7} - 12 x^{3} + 16 x^{7} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.3

Add $- 24 x^{7}$ and $16 x^{7}$ .

$f'' (x) = \frac{12 x^{11} - 8 x^{7} - 12 x^{3} - 8 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.3.4

Subtract $8 x^{3}$ from $- 12 x^{3}$ .

$f'' (x) = \frac{12 x^{11} - 8 x^{7} - 20 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.4

Simplify the numerator.

Tap for more steps...

Step 2.5.4.1

Factor $4 x^{3}$ out of $12 x^{11} - 8 x^{7} - 20 x^{3}$ .

Tap for more steps...

Step 2.5.4.1.1

Factor $4 x^{3}$ out of $12 x^{11}$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8}) - 8 x^{7} - 20 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.1.2

Factor $4 x^{3}$ out of $- 8 x^{7}$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8}) + 4 x^{3} (- 2 x^{4}) - 20 x^{3}}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.1.3

Factor $4 x^{3}$ out of $- 20 x^{3}$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8}) + 4 x^{3} (- 2 x^{4}) + 4 x^{3} (- 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.1.4

Factor $4 x^{3}$ out of $4 x^{3} (3 x^{8}) + 4 x^{3} (- 2 x^{4})$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8} - 2 x^{4}) + 4 x^{3} (- 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.1.5

Factor $4 x^{3}$ out of $4 x^{3} (3 x^{8} - 2 x^{4}) + 4 x^{3} (- 5)$ .

$f'' (x) = \frac{4 x^{3} (3 x^{8} - 2 x^{4} - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$f'' (x) = \frac{4 x^{3} (3 {(x^{4})}^{2} - 2 x^{4} - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.3

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$f'' (x) = \frac{4 x^{3} (3 u^{2} - 2 u - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.4

Factor by grouping.

Tap for more steps...

Step 2.5.4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = - 2$ .

Tap for more steps...

Step 2.5.4.4.1.1

Factor $- 2$ out of $- 2 u$ .

$f'' (x) = \frac{4 x^{3} (3 u^{2} - 2 u - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.4.1.2

Rewrite $- 2$ as $3$ plus $- 5$

$f'' (x) = \frac{4 x^{3} (3 u^{2} + (3 - 5) u - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.4.1.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{3} (3 u^{2} + 3 u - 5 u - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.5.4.4.2.1

Group the first two terms and the last two terms.

$f'' (x) = \frac{4 x^{3} ((3 u^{2} + 3 u) - 5 u - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.4.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = \frac{4 x^{3} (3 u (u + 1) - 5 (u + 1))}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.4.3

Factor the polynomial by factoring out the greatest common factor, $u + 1$ .

$f'' (x) = \frac{4 x^{3} ((u + 1) (3 u - 5))}{{(x^{4} + 1)}^{4}}$

Step 2.5.4.5

Replace all occurrences of $u$ with $x^{4}$ .

$f'' (x) = \frac{4 x^{3} (x^{4} + 1) (3 x^{4} - 5)}{{(x^{4} + 1)}^{4}}$

Step 2.5.5

Cancel the common factor of $x^{4} + 1$ and ${(x^{4} + 1)}^{4}$ .

Tap for more steps...

Step 2.5.5.1

Factor $x^{4} + 1$ out of $4 x^{3} (x^{4} + 1) (3 x^{4} - 5)$ .

$f'' (x) = \frac{(x^{4} + 1) (4 x^{3} (3 x^{4} - 5))}{{(x^{4} + 1)}^{4}}$

Step 2.5.5.2

Cancel the common factors.

Tap for more steps...

Step 2.5.5.2.1

Factor $x^{4} + 1$ out of ${(x^{4} + 1)}^{4}$ .

$f'' (x) = \frac{(x^{4} + 1) (4 x^{3} (3 x^{4} - 5))}{(x^{4} + 1) {(x^{4} + 1)}^{3}}$

Step 2.5.5.2.2

Cancel the common factor.

$f'' (x) = \frac{(x^{4} + 1) (4 x^{3} (3 x^{4} - 5))}{(x^{4} + 1) {(x^{4} + 1)}^{3}}$

Step 2.5.5.2.3

Rewrite the expression.

$f'' (x) = \frac{4 x^{3} (3 x^{4} - 5)}{{(x^{4} + 1)}^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}} = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

$\frac{(1 + x^{4}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 4.1.2

Differentiate.

Tap for more steps...

Step 4.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(1 + x^{4}) \cdot 1 - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 4.1.2.2

Multiply $1 + x^{4}$ by $1$ .

$\frac{1 + x^{4} - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 4.1.2.3

By the Sum Rule, the derivative of $1 + x^{4}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}]$ .

$\frac{1 + x^{4} - x (\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 4.1.2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1 + x^{4} - x (0 + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 4.1.2.5

Add $0$ and $\frac{d}{d x} [x^{4}]$ .

$\frac{1 + x^{4} - x \frac{d}{d x} [x^{4}]}{{(1 + x^{4})}^{2}}$

Step 4.1.2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1 + x^{4} - x (4 x^{3})}{{(1 + x^{4})}^{2}}$

Step 4.1.2.7

Multiply $4$ by $- 1$ .

$\frac{1 + x^{4} - 4 x \cdot x^{3}}{{(1 + x^{4})}^{2}}$

Step 4.1.3

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 4.1.3.1

Move $x^{3}$ .

$\frac{1 + x^{4} - 4 (x^{3} x)}{{(1 + x^{4})}^{2}}$

Step 4.1.3.2

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 4.1.3.2.1

Raise $x$ to the power of $1$ .

$\frac{1 + x^{4} - 4 (x^{3} x^{1})}{{(1 + x^{4})}^{2}}$

Step 4.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 + x^{4} - 4 x^{3 + 1}}{{(1 + x^{4})}^{2}}$

Step 4.1.3.3

Add $3$ and $1$ .

$\frac{1 + x^{4} - 4 x^{4}}{{(1 + x^{4})}^{2}}$

Step 4.1.4

Subtract $4 x^{4}$ from $x^{4}$ .

$\frac{1 - 3 x^{4}}{{(1 + x^{4})}^{2}}$

Step 4.1.5

Reorder terms.

$f' (x) = \frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$ .

$\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}} = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}} = 0$

Step 5.2

Set the numerator equal to zero.

$- 3 x^{4} + 1 = 0$

Step 5.3

Solve the equation for $x$ .

Tap for more steps...

Step 5.3.1

Subtract $1$ from both sides of the equation.

$- 3 x^{4} = - 1$

Step 5.3.2

Divide each term in $- 3 x^{4} = - 1$ by $- 3$ and simplify.

Tap for more steps...

Step 5.3.2.1

Divide each term in $- 3 x^{4} = - 1$ by $- 3$ .

$\frac{- 3 x^{4}}{- 3} = \frac{- 1}{- 3}$

Step 5.3.2.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x^{4}}{- 3} = \frac{- 1}{- 3}$

Step 5.3.2.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 1}{- 3}$

Step 5.3.2.3

Simplify the right side.

Tap for more steps...

Step 5.3.2.3.1

Dividing two negative values results in a positive value.

$x^{4} = \frac{1}{3}$

Step 5.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{\frac{1}{3}}$

Step 5.3.4

Simplify $\pm \sqrt[4]{\frac{1}{3}}$ .

Tap for more steps...

Step 5.3.4.1

Rewrite $\sqrt[4]{\frac{1}{3}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{3}}$ .

$x = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{3}}$

Step 5.3.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[4]{3}}$

Step 5.3.4.3

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$x = \pm \frac{1}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 5.3.4.4

Combine and simplify the denominator.

Tap for more steps...

Step 5.3.4.4.1

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 5.3.4.4.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 5.3.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 5.3.4.4.4

Add $1$ and $3$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 5.3.4.4.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

Tap for more steps...

Step 5.3.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 5.3.4.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 5.3.4.4.5.3

Combine $\frac{1}{4}$ and $4$ .

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 5.3.4.4.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.3.4.4.5.4.1

Cancel the common factor.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 5.3.4.4.5.4.2

Rewrite the expression.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{1}}$

Step 5.3.4.4.5.5

Evaluate the exponent.

$x = \pm \frac{{\sqrt[4]{3}}^{3}}{3}$

Step 5.3.4.5

Simplify the numerator.

Tap for more steps...

Step 5.3.4.5.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$x = \pm \frac{\sqrt[4]{3^{3}}}{3}$

Step 5.3.4.5.2

Raise $3$ to the power of $3$ .

$x = \pm \frac{\sqrt[4]{27}}{3}$

Step 5.3.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 5.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt[4]{27}}{3}$

Step 5.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt[4]{27}}{3}$

Step 5.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 8

Evaluate the second derivative at $x = \frac{\sqrt[4]{27}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 {(\frac{\sqrt[4]{27}}{3})}^{3} (3 {(\frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \frac{{\sqrt[4]{27}}^{3}}{3^{3}} (3 {(\frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.2

Combine $4$ and $\frac{{\sqrt[4]{27}}^{3}}{3^{3}}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 {(\frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.3

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{{\sqrt[4]{27}}^{4}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Tap for more steps...

Step 9.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.4.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.4.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 9.1.4.4.1

Cancel the common factor.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.4.4.2

Rewrite the expression.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27^{1}}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.4.5

Evaluate the exponent.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27}{3^{4}} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.5

Raise $3$ to the power of $4$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 (\frac{27}{81}) - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.1.6.1

Factor $3$ out of $81$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27}{3 (27)} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.6.2

Cancel the common factor.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (3 \frac{27}{3 \cdot 27} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.6.3

Rewrite the expression.

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (\frac{27}{27} - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.7

Divide $27$ by $27$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} (1 - 5)}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.8

Subtract $5$ from $1$ .

$\frac{\frac{4 {\sqrt[4]{27}}^{3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.9

Simplify the numerator.

Tap for more steps...

Step 9.1.9.1

Rewrite ${\sqrt[4]{27}}^{3}$ as $\sqrt[4]{27^{3}}$ .

$\frac{\frac{4 \sqrt[4]{27^{3}}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.9.2

Raise $27$ to the power of $3$ .

$\frac{\frac{4 \sqrt[4]{19683}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.9.3

Rewrite $19683$ as $9^{4} \cdot 3$ .

Tap for more steps...

Step 9.1.9.3.1

Factor $6561$ out of $19683$ .

$\frac{\frac{4 \sqrt[4]{6561 (3)}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.9.3.2

Rewrite $6561$ as $9^{4}$ .

$\frac{\frac{4 \sqrt[4]{9^{4} \cdot 3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.9.4

Pull terms out from under the radical.

$\frac{\frac{4 \cdot 9 \sqrt[4]{3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.9.5

Multiply $4$ by $9$ .

$\frac{\frac{36 \sqrt[4]{3}}{3^{3}} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.10

Raise $3$ to the power of $3$ .

$\frac{\frac{36 \sqrt[4]{3}}{27} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.11

Cancel the common factor of $36$ and $27$ .

Tap for more steps...

Step 9.1.11.1

Factor $9$ out of $36 \sqrt[4]{3}$ .

$\frac{\frac{9 (4 \sqrt[4]{3})}{27} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.11.2

Cancel the common factors.

Tap for more steps...

Step 9.1.11.2.1

Factor $9$ out of $27$ .

$\frac{\frac{9 (4 \sqrt[4]{3})}{9 (3)} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.11.2.2

Cancel the common factor.

$\frac{\frac{9 (4 \sqrt[4]{3})}{9 \cdot 3} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.1.11.2.3

Rewrite the expression.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{({(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 9.2

Simplify the denominator.

Tap for more steps...

Step 9.2.1

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 9.2.2

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Tap for more steps...

Step 9.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} + 1)}^{3}}$

Step 9.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} + 1)}^{3}}$

Step 9.2.2.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 9.2.2.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 9.2.2.4.1

Cancel the common factor.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 9.2.2.4.2

Rewrite the expression.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27^{1}}{3^{4}} + 1)}^{3}}$

Step 9.2.2.5

Evaluate the exponent.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27}{3^{4}} + 1)}^{3}}$

Step 9.2.3

Raise $3$ to the power of $4$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27}{81} + 1)}^{3}}$

Step 9.2.4

Cancel the common factor of $27$ and $81$ .

Tap for more steps...

Step 9.2.4.1

Factor $27$ out of $27$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27 (1)}{81} + 1)}^{3}}$

Step 9.2.4.2

Cancel the common factors.

Tap for more steps...

Step 9.2.4.2.1

Factor $27$ out of $81$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 9.2.4.2.2

Cancel the common factor.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 9.2.4.2.3

Rewrite the expression.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{1}{3} + 1)}^{3}}$

Step 9.2.5

Write $1$ as a fraction with a common denominator.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{1}{3} + \frac{3}{3})}^{3}}$

Step 9.2.6

Combine the numerators over the common denominator.

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{1 + 3}{3})}^{3}}$

Step 9.2.7

Add $1$ and $3$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{{(\frac{4}{3})}^{3}}$

Step 9.2.8

Apply the product rule to $\frac{4}{3}$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{\frac{4^{3}}{3^{3}}}$

Step 9.2.9

Raise $4$ to the power of $3$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{\frac{64}{3^{3}}}$

Step 9.2.10

Raise $3$ to the power of $3$ .

$\frac{\frac{4 \sqrt[4]{3}}{3} \cdot - 4}{\frac{64}{27}}$

Step 9.3

Combine fractions.

Tap for more steps...

Step 9.3.1

Combine $\frac{4 \sqrt[4]{3}}{3}$ and $- 4$ .

$\frac{\frac{4 \sqrt[4]{3} \cdot - 4}{3}}{\frac{64}{27}}$

Step 9.3.2

Simplify the expression.

Tap for more steps...

Step 9.3.2.1

Multiply $- 4$ by $4$ .

$\frac{\frac{- 16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 9.3.2.2

Move the negative in front of the fraction.

$\frac{- \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{16 \sqrt[4]{3}}{3} \cdot \frac{27}{64}$

Step 9.5

Cancel the common factor of $16$ .

Tap for more steps...

Step 9.5.1

Move the leading negative in $- \frac{16 \sqrt[4]{3}}{3}$ into the numerator.

$\frac{- 16 \sqrt[4]{3}}{3} \cdot \frac{27}{64}$

Step 9.5.2

Factor $16$ out of $- 16 \sqrt[4]{3}$ .

$\frac{16 (- \sqrt[4]{3})}{3} \cdot \frac{27}{64}$

Step 9.5.3

Factor $16$ out of $64$ .

$\frac{16 (- \sqrt[4]{3})}{3} \cdot \frac{27}{16 (4)}$

Step 9.5.4

Cancel the common factor.

$\frac{16 (- \sqrt[4]{3})}{3} \cdot \frac{27}{16 \cdot 4}$

Step 9.5.5

Rewrite the expression.

$\frac{- \sqrt[4]{3}}{3} \cdot \frac{27}{4}$

Step 9.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.6.1

Factor $3$ out of $27$ .

$\frac{- \sqrt[4]{3}}{3} \cdot \frac{3 (9)}{4}$

Step 9.6.2

Cancel the common factor.

$\frac{- \sqrt[4]{3}}{3} \cdot \frac{3 \cdot 9}{4}$

Step 9.6.3

Rewrite the expression.

$- \sqrt[4]{3} \frac{9}{4}$

Step 9.7

Combine $\frac{9}{4}$ and $\sqrt[4]{3}$ .

$- \frac{9 \sqrt[4]{3}}{4}$

Step 10

$x = \frac{\sqrt[4]{27}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{\sqrt[4]{27}}{3}$ is a local maximum

Step 11

Find the y-value when $x = \frac{\sqrt[4]{27}}{3}$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $\frac{\sqrt[4]{27}}{3}$ in the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\frac{\sqrt[4]{27}}{3}}{1 + {(\frac{\sqrt[4]{27}}{3})}^{4}}$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(\frac{\sqrt[4]{27}}{3})}^{4}}$

Step 11.2.2

Simplify the denominator.

Tap for more steps...

Step 11.2.2.1

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{\sqrt[4]{27}}^{4}}{3^{4}}}$

Step 11.2.2.2

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Tap for more steps...

Step 11.2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}}}$

Step 11.2.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}}}$

Step 11.2.2.2.3

Combine $\frac{1}{4}$ and $4$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 11.2.2.2.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 11.2.2.2.4.1

Cancel the common factor.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 11.2.2.2.4.2

Rewrite the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 11.2.2.2.5

Evaluate the exponent.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 11.2.2.3

Raise $3$ to the power of $4$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{81}}$

Step 11.2.2.4

Cancel the common factor of $27$ and $81$ .

Tap for more steps...

Step 11.2.2.4.1

Factor $27$ out of $27$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 (1)}{81}}$

Step 11.2.2.4.2

Cancel the common factors.

Tap for more steps...

Step 11.2.2.4.2.1

Factor $27$ out of $81$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 11.2.2.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 11.2.2.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{1}{3}}$

Step 11.2.2.5

Write $1$ as a fraction with a common denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3}{3} + \frac{1}{3}}$

Step 11.2.2.6

Combine the numerators over the common denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3 + 1}{3}}$

Step 11.2.2.7

Add $3$ and $1$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{4}{3}}$

Step 11.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot (1 (\frac{3}{4}))$

Step 11.2.4

Multiply $\frac{3}{4}$ by $1$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 11.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.2.5.1

Cancel the common factor.

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 11.2.5.2

Rewrite the expression.

$f (\frac{\sqrt[4]{27}}{3}) = \sqrt[4]{27} (\frac{1}{4})$

Step 11.2.6

Combine $\sqrt[4]{27}$ and $\frac{1}{4}$ .

$f (\frac{\sqrt[4]{27}}{3}) = \frac{\sqrt[4]{27}}{4}$

Step 11.2.7

The final answer is $\frac{\sqrt[4]{27}}{4}$ .

$y = \frac{\sqrt[4]{27}}{4}$

Step 12

Evaluate the second derivative at $x = - \frac{\sqrt[4]{27}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 {(- \frac{\sqrt[4]{27}}{3})}^{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify the numerator.

Tap for more steps...

Step 13.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$\frac{4 {(- 1)}^{3} {(\frac{\sqrt[4]{27}}{3})}^{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.2

Raise $- 1$ to the power of $3$ .

$\frac{4 \cdot - 1 {(\frac{\sqrt[4]{27}}{3})}^{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.3

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \cdot - 1 \frac{{\sqrt[4]{27}}^{3}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.4

Simplify the numerator.

Tap for more steps...

Step 13.1.4.1

Rewrite ${\sqrt[4]{27}}^{3}$ as $\sqrt[4]{27^{3}}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{27^{3}}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.4.2

Raise $27$ to the power of $3$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{19683}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.4.3

Rewrite $19683$ as $9^{4} \cdot 3$ .

Tap for more steps...

Step 13.1.4.3.1

Factor $6561$ out of $19683$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{6561 (3)}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.4.3.2

Rewrite $6561$ as $9^{4}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{9^{4} \cdot 3}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.4.4

Pull terms out from under the radical.

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{3^{3}} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.5

Raise $3$ to the power of $3$ .

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{27} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.6

Cancel the common factor of $9$ and $27$ .

Tap for more steps...

Step 13.1.6.1

Factor $9$ out of $9 \sqrt[4]{3}$ .

$\frac{4 \cdot - 1 \frac{9 (\sqrt[4]{3})}{27} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.6.2

Cancel the common factors.

Tap for more steps...

Step 13.1.6.2.1

Factor $9$ out of $27$ .

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{9 \cdot 3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.6.2.2

Cancel the common factor.

$\frac{4 \cdot - 1 \frac{9 \sqrt[4]{3}}{9 \cdot 3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.6.2.3

Rewrite the expression.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 {(- \frac{\sqrt[4]{27}}{3})}^{4} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7

Simplify each term.

Tap for more steps...

Step 13.1.7.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 13.1.7.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 ({(- 1)}^{4} {(\frac{\sqrt[4]{27}}{3})}^{4}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.1.2

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 ({(- 1)}^{4} \frac{{\sqrt[4]{27}}^{4}}{3^{4}}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.2

Raise $- 1$ to the power of $4$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 (1 \frac{{\sqrt[4]{27}}^{4}}{3^{4}}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.3

Multiply $\frac{{\sqrt[4]{27}}^{4}}{3^{4}}$ by $1$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{{\sqrt[4]{27}}^{4}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Tap for more steps...

Step 13.1.7.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.4.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.4.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 13.1.7.4.4.1

Cancel the common factor.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{\frac{4}{4}}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.4.4.2

Rewrite the expression.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27^{1}}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.4.5

Evaluate the exponent.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27}{3^{4}} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.5

Raise $3$ to the power of $4$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 (\frac{27}{81}) - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.1.7.6.1

Factor $3$ out of $81$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27}{3 (27)} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.6.2

Cancel the common factor.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (3 \frac{27}{3 \cdot 27} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.6.3

Rewrite the expression.

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (\frac{27}{27} - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.7.7

Divide $27$ by $27$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} (1 - 5)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.8

Subtract $5$ from $1$ .

$\frac{4 \cdot - 1 \frac{\sqrt[4]{3}}{3} \cdot - 4}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.9

Combine exponents.

Tap for more steps...

Step 13.1.9.1

Factor out negative.

$\frac{- (4 \frac{\sqrt[4]{3}}{3} \cdot - 4)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.9.2

Combine $4$ and $\frac{\sqrt[4]{3}}{3}$ .

$\frac{- (\frac{4 \sqrt[4]{3}}{3} \cdot - 4)}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.9.3

Combine $\frac{4 \sqrt[4]{3}}{3}$ and $- 4$ .

$\frac{- \frac{4 \sqrt[4]{3} \cdot - 4}{3}}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.9.4

Multiply $- 4$ by $4$ .

$\frac{- \frac{- 16 \sqrt[4]{3}}{3}}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.1.10

Move the negative in front of the fraction.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{({(- \frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.2

Simplify the denominator.

Tap for more steps...

Step 13.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 13.2.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{({(- 1)}^{4} {(\frac{\sqrt[4]{27}}{3})}^{4} + 1)}^{3}}$

Step 13.2.1.2

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{({(- 1)}^{4} \frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 13.2.2

Raise $- 1$ to the power of $4$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(1 \frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 13.2.3

Multiply $\frac{{\sqrt[4]{27}}^{4}}{3^{4}}$ by $1$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{{\sqrt[4]{27}}^{4}}{3^{4}} + 1)}^{3}}$

Step 13.2.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Tap for more steps...

Step 13.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}} + 1)}^{3}}$

Step 13.2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{\frac{1}{4} \cdot 4}}{3^{4}} + 1)}^{3}}$

Step 13.2.4.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 13.2.4.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 13.2.4.4.1

Cancel the common factor.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{\frac{4}{4}}}{3^{4}} + 1)}^{3}}$

Step 13.2.4.4.2

Rewrite the expression.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27^{1}}{3^{4}} + 1)}^{3}}$

Step 13.2.4.5

Evaluate the exponent.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27}{3^{4}} + 1)}^{3}}$

Step 13.2.5

Raise $3$ to the power of $4$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27}{81} + 1)}^{3}}$

Step 13.2.6

Cancel the common factor of $27$ and $81$ .

Tap for more steps...

Step 13.2.6.1

Factor $27$ out of $27$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27 (1)}{81} + 1)}^{3}}$

Step 13.2.6.2

Cancel the common factors.

Tap for more steps...

Step 13.2.6.2.1

Factor $27$ out of $81$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 13.2.6.2.2

Cancel the common factor.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{27 \cdot 1}{27 \cdot 3} + 1)}^{3}}$

Step 13.2.6.2.3

Rewrite the expression.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{1}{3} + 1)}^{3}}$

Step 13.2.7

Write $1$ as a fraction with a common denominator.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{1}{3} + \frac{3}{3})}^{3}}$

Step 13.2.8

Combine the numerators over the common denominator.

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{1 + 3}{3})}^{3}}$

Step 13.2.9

Add $1$ and $3$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{{(\frac{4}{3})}^{3}}$

Step 13.2.10

Apply the product rule to $\frac{4}{3}$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{\frac{4^{3}}{3^{3}}}$

Step 13.2.11

Raise $4$ to the power of $3$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{3^{3}}}$

Step 13.2.12

Raise $3$ to the power of $3$ .

$\frac{- - \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 13.3

Simplify the numerator.

Tap for more steps...

Step 13.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1 \frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 13.3.2

Multiply $\frac{16 \sqrt[4]{3}}{3}$ by $1$ .

$\frac{\frac{16 \sqrt[4]{3}}{3}}{\frac{64}{27}}$

Step 13.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 \sqrt[4]{3}}{3} \cdot \frac{27}{64}$

Step 13.5

Cancel the common factor of $16$ .

Tap for more steps...

Step 13.5.1

Factor $16$ out of $16 \sqrt[4]{3}$ .

$\frac{16 (\sqrt[4]{3})}{3} \cdot \frac{27}{64}$

Step 13.5.2

Factor $16$ out of $64$ .

$\frac{16 (\sqrt[4]{3})}{3} \cdot \frac{27}{16 (4)}$

Step 13.5.3

Cancel the common factor.

$\frac{16 \sqrt[4]{3}}{3} \cdot \frac{27}{16 \cdot 4}$

Step 13.5.4

Rewrite the expression.

$\frac{\sqrt[4]{3}}{3} \cdot \frac{27}{4}$

Step 13.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 13.6.1

Factor $3$ out of $27$ .

$\frac{\sqrt[4]{3}}{3} \cdot \frac{3 (9)}{4}$

Step 13.6.2

Cancel the common factor.

$\frac{\sqrt[4]{3}}{3} \cdot \frac{3 \cdot 9}{4}$

Step 13.6.3

Rewrite the expression.

$\sqrt[4]{3} \frac{9}{4}$

Step 13.7

Combine $\sqrt[4]{3}$ and $\frac{9}{4}$ .

$\frac{\sqrt[4]{3} \cdot 9}{4}$

Step 13.8

Move $9$ to the left of $\sqrt[4]{3}$ .

$\frac{9 \sqrt[4]{3}}{4}$

Step 14

$x = - \frac{\sqrt[4]{27}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{\sqrt[4]{27}}{3}$ is a local minimum

Step 15

Find the y-value when $x = - \frac{\sqrt[4]{27}}{3}$ .

Tap for more steps...

Step 15.1

Replace the variable $x$ with $- \frac{\sqrt[4]{27}}{3}$ in the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = \frac{- \frac{\sqrt[4]{27}}{3}}{1 + {(- \frac{\sqrt[4]{27}}{3})}^{4}}$

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(- \frac{\sqrt[4]{27}}{3})}^{4}}$

Step 15.2.2

Simplify the denominator.

Tap for more steps...

Step 15.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 15.2.2.1.1

Apply the product rule to $- \frac{\sqrt[4]{27}}{3}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(- 1)}^{4} {(\frac{\sqrt[4]{27}}{3})}^{4}}$

Step 15.2.2.1.2

Apply the product rule to $\frac{\sqrt[4]{27}}{3}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + {(- 1)}^{4} (\frac{{\sqrt[4]{27}}^{4}}{3^{4}})}$

Step 15.2.2.2

Raise $- 1$ to the power of $4$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + 1 (\frac{{\sqrt[4]{27}}^{4}}{3^{4}})}$

Step 15.2.2.3

Multiply $\frac{{\sqrt[4]{27}}^{4}}{3^{4}}$ by $1$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{\sqrt[4]{27}}^{4}}{3^{4}}}$

Step 15.2.2.4

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Tap for more steps...

Step 15.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{{(27^{\frac{1}{4}})}^{4}}{3^{4}}}$

Step 15.2.2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{1}{4} \cdot 4}}{3^{4}}}$

Step 15.2.2.4.3

Combine $\frac{1}{4}$ and $4$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 15.2.2.4.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 15.2.2.4.4.1

Cancel the common factor.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27^{\frac{4}{4}}}{3^{4}}}$

Step 15.2.2.4.4.2

Rewrite the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 15.2.2.4.5

Evaluate the exponent.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{3^{4}}}$

Step 15.2.2.5

Raise $3$ to the power of $4$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27}{81}}$

Step 15.2.2.6

Cancel the common factor of $27$ and $81$ .

Tap for more steps...

Step 15.2.2.6.1

Factor $27$ out of $27$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 (1)}{81}}$

Step 15.2.2.6.2

Cancel the common factors.

Tap for more steps...

Step 15.2.2.6.2.1

Factor $27$ out of $81$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 15.2.2.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{27 \cdot 1}{27 \cdot 3}}$

Step 15.2.2.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{1 + \frac{1}{3}}$

Step 15.2.2.7

Write $1$ as a fraction with a common denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3}{3} + \frac{1}{3}}$

Step 15.2.2.8

Combine the numerators over the common denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{3 + 1}{3}}$

Step 15.2.2.9

Add $3$ and $1$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{1}{\frac{4}{3}}$

Step 15.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot (1 (\frac{3}{4}))$

Step 15.2.4

Multiply $\frac{3}{4}$ by $1$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 15.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.2.5.1

Move the leading negative in $- \frac{\sqrt[4]{27}}{3}$ into the numerator.

$f (- \frac{\sqrt[4]{27}}{3}) = \frac{- \sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 15.2.5.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{27}}{3}) = \frac{- \sqrt[4]{27}}{3} \cdot \frac{3}{4}$

Step 15.2.5.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{27}}{3}) = - \sqrt[4]{27} \frac{1}{4}$

Step 15.2.6

Combine $\frac{1}{4}$ and $\sqrt[4]{27}$ .

$f (- \frac{\sqrt[4]{27}}{3}) = - \frac{\sqrt[4]{27}}{4}$

Step 15.2.7

The final answer is $- \frac{\sqrt[4]{27}}{4}$ .

$y = - \frac{\sqrt[4]{27}}{4}$

Step 16

These are the local extrema for $f (x) = \frac{x}{1 + x^{4}}$ .

$(\frac{\sqrt[4]{27}}{3}, \frac{\sqrt[4]{27}}{4})$ is a local maxima

$(- \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{4})$ is a local minima

Step 17