Calculus Examples

$f (x) = \frac{x}{x^{2} - 1}$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 1.1

Find the second derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

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 $g (x) = x^{2} - 1$ .

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

Step 1.1.1.2

Differentiate.

Tap for more steps...

Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Multiply $x^{2} - 1$ by $1$ .

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

Step 1.1.1.2.3

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Simplify the expression.

Tap for more steps...

Step 1.1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

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

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

Step 1.1.1.6

Add $1$ and $1$ .

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

Step 1.1.1.7

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 1.1.2

Find the second derivative.

Tap for more steps...

Step 1.1.2.1

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

Step 1.1.2.2

Differentiate.

Tap for more steps...

Step 1.1.2.2.1

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

Tap for more steps...

Step 1.1.2.2.1.1

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

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

Step 1.1.2.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.2.2.2

By the Sum Rule, the derivative of $- x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.2.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 1.1.2.2.4

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

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

Step 1.1.2.2.5

Multiply $2$ by $- 1$ .

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

Step 1.1.2.2.6

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

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

Step 1.1.2.2.7

Add $- 2 x$ and $0$ .

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

Step 1.1.2.3

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 $g (x) = x^{2} - 1$ .

Tap for more steps...

Step 1.1.2.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

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

Step 1.1.2.4

Differentiate.

Tap for more steps...

Step 1.1.2.4.1

Multiply $2$ by $- 1$ .

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

Step 1.1.2.4.2

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.2.4.3

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

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

Step 1.1.2.4.4

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

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

Step 1.1.2.4.5

Simplify the expression.

Tap for more steps...

Step 1.1.2.4.5.1

Add $2 x$ and $0$ .

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

Step 1.1.2.4.5.2

Move $2$ to the left of $x^{2} - 1$ .

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

Step 1.1.2.4.5.3

Multiply $2$ by $- 2$ .

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

Step 1.1.2.5

Simplify.

Tap for more steps...

Step 1.1.2.5.1

Apply the distributive property.

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

Step 1.1.2.5.2

Apply the distributive property.

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

Step 1.1.2.5.3

Simplify the numerator.

Tap for more steps...

Step 1.1.2.5.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.5.3.1.2

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

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

Step 1.1.2.5.3.1.3

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.1.2.5.3.1.3.1

Apply the distributive property.

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

Step 1.1.2.5.3.1.3.2

Apply the distributive property.

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

Step 1.1.2.5.3.1.3.3

Apply the distributive property.

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

Step 1.1.2.5.3.1.4

Simplify and combine like terms.

Tap for more steps...

Step 1.1.2.5.3.1.4.1

Simplify each term.

Tap for more steps...

Step 1.1.2.5.3.1.4.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.1.2.5.3.1.4.1.1.1

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

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

Step 1.1.2.5.3.1.4.1.1.2

Add $2$ and $2$ .

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

Step 1.1.2.5.3.1.4.1.2

Move $- 1$ to the left of $x^{2}$ .

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

Step 1.1.2.5.3.1.4.1.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 1.1.2.5.3.1.4.1.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 1.1.2.5.3.1.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.1.2.5.3.1.4.2

Subtract $x^{2}$ from $- x^{2}$ .

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

Step 1.1.2.5.3.1.5

Apply the distributive property.

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

Step 1.1.2.5.3.1.6

Simplify.

Tap for more steps...

Step 1.1.2.5.3.1.6.1

Multiply $- 2$ by $- 2$ .

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

Step 1.1.2.5.3.1.6.2

Multiply $- 2$ by $1$ .

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

Step 1.1.2.5.3.1.7

Apply the distributive property.

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

Step 1.1.2.5.3.1.8

Simplify.

Tap for more steps...

Step 1.1.2.5.3.1.8.1

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

Tap for more steps...

Step 1.1.2.5.3.1.8.1.1

Move $x$ .

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

Step 1.1.2.5.3.1.8.1.2

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

Tap for more steps...

Step 1.1.2.5.3.1.8.1.2.1

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

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

Step 1.1.2.5.3.1.8.1.2.2

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

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

Step 1.1.2.5.3.1.8.1.3

Add $1$ and $4$ .

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

Step 1.1.2.5.3.1.8.2

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.2.5.3.1.8.2.1

Move $x$ .

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

Step 1.1.2.5.3.1.8.2.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.1.2.5.3.1.8.2.2.1

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

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

Step 1.1.2.5.3.1.8.2.2.2

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

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

Step 1.1.2.5.3.1.8.2.3

Add $1$ and $2$ .

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

Step 1.1.2.5.3.1.9

Simplify each term.

Tap for more steps...

Step 1.1.2.5.3.1.9.1

Multiply $- 1$ by $- 4$ .

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

Step 1.1.2.5.3.1.9.2

Multiply $- 4$ by $- 1$ .

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

Step 1.1.2.5.3.1.10

Simplify each term.

Tap for more steps...

Step 1.1.2.5.3.1.10.1

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.2.5.3.1.10.1.1

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 1.1.2.5.3.1.10.1.1.1

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

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

Step 1.1.2.5.3.1.10.1.1.2

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

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

Step 1.1.2.5.3.1.10.1.2

Add $2$ and $1$ .

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

Step 1.1.2.5.3.1.10.2

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.2.5.3.1.11

Expand $(4 x^{2} + 4) (x^{3} - x)$ using the FOIL Method.

Tap for more steps...

Step 1.1.2.5.3.1.11.1

Apply the distributive property.

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

Step 1.1.2.5.3.1.11.2

Apply the distributive property.

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

Step 1.1.2.5.3.1.11.3

Apply the distributive property.

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

Step 1.1.2.5.3.1.12

Simplify and combine like terms.

Tap for more steps...

Step 1.1.2.5.3.1.12.1

Simplify each term.

Tap for more steps...

Step 1.1.2.5.3.1.12.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 1.1.2.5.3.1.12.1.1.1

Move $x^{3}$ .

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

Step 1.1.2.5.3.1.12.1.1.2

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

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

Step 1.1.2.5.3.1.12.1.1.3

Add $3$ and $2$ .

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

Step 1.1.2.5.3.1.12.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.5.3.1.12.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.2.5.3.1.12.1.3.1

Move $x$ .

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

Step 1.1.2.5.3.1.12.1.3.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.1.2.5.3.1.12.1.3.2.1

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

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

Step 1.1.2.5.3.1.12.1.3.2.2

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

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

Step 1.1.2.5.3.1.12.1.3.3

Add $1$ and $2$ .

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

Step 1.1.2.5.3.1.12.1.4

Multiply $4$ by $- 1$ .

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

Step 1.1.2.5.3.1.12.1.5

Multiply $- 1$ by $4$ .

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

Step 1.1.2.5.3.1.12.2

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

$f'' (x) = \frac{- 2 x^{5} + 4 x^{3} - 2 x + 4 x^{5} + 0 - 4 x}{{(x^{2} - 1)}^{4}}$

Step 1.1.2.5.3.1.12.3

Add $4 x^{5}$ and $0$ .

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

Step 1.1.2.5.3.2

Add $- 2 x^{5}$ and $4 x^{5}$ .

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

Step 1.1.2.5.3.3

Subtract $4 x$ from $- 2 x$ .

$f'' (x) = \frac{2 x^{5} + 4 x^{3} - 6 x}{{(x^{2} - 1)}^{4}}$

Step 1.1.2.5.4

Simplify the numerator.

Tap for more steps...

Step 1.1.2.5.4.1

Factor $2 x$ out of $2 x^{5} + 4 x^{3} - 6 x$ .

Tap for more steps...

Step 1.1.2.5.4.1.1

Factor $2 x$ out of $2 x^{5}$ .

$f'' (x) = \frac{2 x (x^{4}) + 4 x^{3} - 6 x}{{(x^{2} - 1)}^{4}}$

Step 1.1.2.5.4.1.2

Factor $2 x$ out of $4 x^{3}$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (2 x^{2}) - 6 x}{{(x^{2} - 1)}^{4}}$

Step 1.1.2.5.4.1.3

Factor $2 x$ out of $- 6 x$ .

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

Step 1.1.2.5.4.1.4

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

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

Step 1.1.2.5.4.1.5

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

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

Step 1.1.2.5.4.2

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

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

Step 1.1.2.5.4.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 1.1.2.5.4.4

Factor $u^{2} + 2 u - 3$ using the AC method.

Tap for more steps...

Step 1.1.2.5.4.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.1.2.5.4.4.2

Write the factored form using these integers.

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

Step 1.1.2.5.4.5

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 1.1.2.5.4.6

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.2.5.4.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.1.2.5.5

Simplify the denominator.

Tap for more steps...

Step 1.1.2.5.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.2.5.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.1.2.5.5.3

Apply the product rule to $(x + 1) (x - 1)$ .

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

Step 1.1.2.5.6

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

Tap for more steps...

Step 1.1.2.5.6.1

Factor $x + 1$ out of $2 x (x + 1) (x - 1) (x^{2} + 3)$ .

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

Step 1.1.2.5.6.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.5.6.2.1

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

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

Step 1.1.2.5.6.2.2

Cancel the common factor.

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

Step 1.1.2.5.6.2.3

Rewrite the expression.

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

Step 1.1.2.5.7

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

Tap for more steps...

Step 1.1.2.5.7.1

Factor $x - 1$ out of $(2 x (x - 1)) (x^{2} + 3)$ .

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

Step 1.1.2.5.7.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.5.7.2.1

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

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

Step 1.1.2.5.7.2.2

Cancel the common factor.

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

Step 1.1.2.5.7.2.3

Rewrite the expression.

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

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

Step 1.1.3

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Set the second derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$2 x (x^{2} + 3) = 0$

Step 1.2.3

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} + 3 = 0$

Step 1.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.3

Set $x^{2} + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.3.3.1

Set $x^{2} + 3$ equal to $0$ .

$x^{2} + 3 = 0$

Step 1.2.3.3.2

Solve $x^{2} + 3 = 0$ for $x$ .

Tap for more steps...

Step 1.2.3.3.2.1

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

Step 1.2.3.3.2.2

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

$x = \pm \sqrt{- 3}$

Step 1.2.3.3.2.3

Simplify $\pm \sqrt{- 3}$ .

Tap for more steps...

Step 1.2.3.3.2.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 1.2.3.3.2.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 1.2.3.3.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{3}$

Step 1.2.3.3.2.4

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

Tap for more steps...

Step 1.2.3.3.2.4.1

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

$x = i \sqrt{3}$

Step 1.2.3.3.2.4.2

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

$x = - i \sqrt{3}$

Step 1.2.3.3.2.4.3

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

$x = i \sqrt{3}, - i \sqrt{3}$

Step 1.2.3.4

The final solution is all the values that make $2 x (x^{2} + 3) = 0$ true.

$x = 0, i \sqrt{3}, - i \sqrt{3}$

Step 2

Find the domain of $f (x) = \frac{x}{x^{2} - 1}$ .

Tap for more steps...

Step 2.1

Set the denominator in $\frac{x}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 1 = 0$

Step 2.2

Solve for $x$ .

Tap for more steps...

Step 2.2.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 2.2.2

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

$x = \pm \sqrt{1}$

Step 2.2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.2.4

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

Tap for more steps...

Step 2.2.4.1

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

$x = 1$

Step 2.2.4.2

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

$x = - 1$

Step 2.2.4.3

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

$x = 1, - 1$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 1}$

Interval Notation:

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 1}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 1)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 4.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Multiply $2$ by $- 4$ .

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

Step 4.2.2

Simplify the denominator.

Tap for more steps...

Step 4.2.2.1

Add $- 4$ and $1$ .

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

Step 4.2.2.2

Subtract $1$ from $- 4$ .

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

Step 4.2.2.3

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

$f'' (- 4) = \frac{- 8 ({(- 4)}^{2} + 3)}{- 27 {(- 5)}^{3}}$

Step 4.2.2.4

Raise $- 5$ to the power of $3$ .

$f'' (- 4) = \frac{- 8 ({(- 4)}^{2} + 3)}{- 27 \cdot - 125}$

Step 4.2.3

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1

Raise $- 4$ to the power of $2$ .

$f'' (- 4) = \frac{- 8 (16 + 3)}{- 27 \cdot - 125}$

Step 4.2.3.2

Add $16$ and $3$ .

$f'' (- 4) = \frac{- 8 \cdot 19}{- 27 \cdot - 125}$

Step 4.2.4

Simplify the expression.

Tap for more steps...

Step 4.2.4.1

Multiply $- 27$ by $- 125$ .

$f'' (- 4) = \frac{- 8 \cdot 19}{3375}$

Step 4.2.4.2

Multiply $- 8$ by $19$ .

$f'' (- 4) = \frac{- 152}{3375}$

Step 4.2.4.3

Move the negative in front of the fraction.

$f'' (- 4) = - \frac{152}{3375}$

Step 4.2.5

The final answer is $- \frac{152}{3375}$ .

$- \frac{152}{3375}$

Step 4.3

The graph is concave down on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is negative.

Concave down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- 1, 0)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f'' (- 0.5) = \frac{2 (- 0.5) ({(- 0.5)}^{2} + 3)}{{((- 0.5) + 1)}^{3} {((- 0.5) - 1)}^{3}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Multiply $2$ by $- 0.5$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{(- 0.5 + 1)}^{3} {(- 0.5 - 1)}^{3}}$

Step 5.2.2

Simplify the denominator.

Tap for more steps...

Step 5.2.2.1

Add $- 0.5$ and $1$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{0.5}^{3} {(- 0.5 - 1)}^{3}}$

Step 5.2.2.2

Subtract $1$ from $- 0.5$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{{0.5}^{3} {(- 1.5)}^{3}}$

Step 5.2.2.3

Raise $0.5$ to the power of $3$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{0.125 {(- 1.5)}^{3}}$

Step 5.2.2.4

Raise $- 1.5$ to the power of $3$ .

$f'' (- 0.5) = \frac{- 1 ({(- 0.5)}^{2} + 3)}{0.125 \cdot - 3.375}$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Raise $- 0.5$ to the power of $2$ .

$f'' (- 0.5) = \frac{- 1 (0.25 + 3)}{0.125 \cdot - 3.375}$

Step 5.2.3.2

Add $0.25$ and $3$ .

$f'' (- 0.5) = \frac{- 1 \cdot 3.25}{0.125 \cdot - 3.375}$

Step 5.2.4

Simplify the expression.

Tap for more steps...

Step 5.2.4.1

Multiply $0.125$ by $- 3.375$ .

$f'' (- 0.5) = \frac{- 1 \cdot 3.25}{- 0.421875}$

Step 5.2.4.2

Multiply $- 1$ by $3.25$ .

$f'' (- 0.5) = \frac{- 3.25}{- 0.421875}$

Step 5.2.4.3

Divide $- 3.25$ by $- 0.421875$ .

$f'' (- 0.5) = 7.\overline{703}$

Step 5.2.5

The final answer is $7.\overline{703}$ .

$7.\overline{703}$

Step 5.3

The graph is concave up on the interval $(- 1, 0)$ because $f'' (- 0.5)$ is positive.

Concave up on $(- 1, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, 1)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $0.5$ in the expression.

$f'' (0.5) = \frac{2 (0.5) ({(0.5)}^{2} + 3)}{{((0.5) + 1)}^{3} {((0.5) - 1)}^{3}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify the numerator.

Tap for more steps...

Step 6.2.1.1

Multiply $2$ by $0.5$ .

$f'' (0.5) = \frac{1 ({0.5}^{2} + 3)}{{(0.5 + 1)}^{3} {(0.5 - 1)}^{3}}$

Step 6.2.1.2

Multiply ${0.5}^{2} + 3$ by $1$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{(0.5 + 1)}^{3} {(0.5 - 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

Add $0.5$ and $1$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{1.5}^{3} {(0.5 - 1)}^{3}}$

Step 6.2.2.2

Subtract $1$ from $0.5$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{{1.5}^{3} {(- 0.5)}^{3}}$

Step 6.2.2.3

Raise $1.5$ to the power of $3$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{3.375 {(- 0.5)}^{3}}$

Step 6.2.2.4

Raise $- 0.5$ to the power of $3$ .

$f'' (0.5) = \frac{{0.5}^{2} + 3}{3.375 \cdot - 0.125}$

Step 6.2.3

Simplify the numerator.

Tap for more steps...

Step 6.2.3.1

Raise $0.5$ to the power of $2$ .

$f'' (0.5) = \frac{0.25 + 3}{3.375 \cdot - 0.125}$

Step 6.2.3.2

Add $0.25$ and $3$ .

$f'' (0.5) = \frac{3.25}{3.375 \cdot - 0.125}$

Step 6.2.4

Simplify the expression.

Tap for more steps...

Step 6.2.4.1

Multiply $3.375$ by $- 0.125$ .

$f'' (0.5) = \frac{3.25}{- 0.421875}$

Step 6.2.4.2

Divide $3.25$ by $- 0.421875$ .

$f'' (0.5) = - 7.\overline{703}$

Step 6.2.5

The final answer is $- 7.\overline{703}$ .

$- 7.\overline{703}$

Step 6.3

The graph is concave down on the interval $(0, 1)$ because $f'' (0.5)$ is negative.

Concave down on $(0, 1)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $4$ in the expression.

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

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Multiply $2$ by $4$ .

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

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

Add $4$ and $1$ .

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

Step 7.2.2.2

Subtract $1$ from $4$ .

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

Step 7.2.2.3

Raise $5$ to the power of $3$ .

$f'' (4) = \frac{8 (4^{2} + 3)}{125 \cdot 3^{3}}$

Step 7.2.2.4

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

$f'' (4) = \frac{8 (4^{2} + 3)}{125 \cdot 27}$

Step 7.2.3

Simplify the numerator.

Tap for more steps...

Step 7.2.3.1

Raise $4$ to the power of $2$ .

$f'' (4) = \frac{8 (16 + 3)}{125 \cdot 27}$

Step 7.2.3.2

Add $16$ and $3$ .

$f'' (4) = \frac{8 \cdot 19}{125 \cdot 27}$

Step 7.2.4

Multiply.

Tap for more steps...

Step 7.2.4.1

Multiply $125$ by $27$ .

$f'' (4) = \frac{8 \cdot 19}{3375}$

Step 7.2.4.2

Multiply $8$ by $19$ .

$f'' (4) = \frac{152}{3375}$

Step 7.2.5

The final answer is $\frac{152}{3375}$ .

$\frac{152}{3375}$

Step 7.3

The graph is concave up on the interval $(1, \infty)$ because $f'' (4)$ is positive.

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, - 1)$ since $f'' (x)$ is negative

Concave up on $(- 1, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 1)$ since $f'' (x)$ is negative

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 9