Calculus Examples

$y = \arctan (4 x^{2} - 3)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\arctan (4 x^{2} - 3))$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \arctan (y)$ and $g (y) = 4 x^{2} - 3$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $4 x^{2} - 3$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d y} [4 x^{2} - 3]$

Step 3.1.2

The derivative of $\arctan (u_{1})$ with respect to $u_{1}$ is $\frac{1}{1 + {u_{1}}^{2}}$ .

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d y} [4 x^{2} - 3]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $4 x^{2} - 3$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} \frac{d}{d y} [4 x^{2} - 3]$

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

By the Sum Rule, the derivative of $4 x^{2} - 3$ with respect to $y$ is $\frac{d}{d y} [4 x^{2}] + \frac{d}{d y} [- 3]$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (\frac{d}{d y} [4 x^{2}] + \frac{d}{d y} [- 3])$

Step 3.2.2

Since $4$ is constant with respect to $y$ , the derivative of $4 x^{2}$ with respect to $y$ is $4 \frac{d}{d y} [x^{2}]$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (4 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 3])$

Step 3.3

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

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (4 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- 3])$

Step 3.3.2

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

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (4 (2 u_{2} \frac{d}{d y} [x]) + \frac{d}{d y} [- 3])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $x$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (4 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [- 3])$

Step 3.4

Multiply $2$ by $4$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (8 (x \frac{d}{d y} [x]) + \frac{d}{d y} [- 3])$

Step 3.5

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (8 x x' + \frac{d}{d y} [- 3])$

Step 3.6

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3$ with respect to $y$ is $0$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (8 x x' + 0)$

Step 3.7

Combine fractions.

Tap for more steps...

Step 3.7.1

Add $8 x x'$ and $0$ .

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

Step 3.7.2

Combine $8$ and $\frac{1}{1 + {(4 x^{2} - 3)}^{2}}$ .

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

Step 3.7.3

Combine $x$ and $\frac{8}{1 + {(4 x^{2} - 3)}^{2}}$ .

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

Step 3.7.4

Combine $\frac{x \cdot 8}{1 + {(4 x^{2} - 3)}^{2}}$ and $x'$ .

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

Step 3.7.5

Move $8$ to the left of $x$ .

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

Step 3.8

Simplify.

Tap for more steps...

Step 3.8.1

Reorder terms.

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

Step 3.8.2

Simplify the denominator.

Tap for more steps...

Step 3.8.2.1

Rewrite ${(4 x^{2} - 3)}^{2}$ as $(4 x^{2} - 3) (4 x^{2} - 3)$ .

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

Step 3.8.2.2

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

Tap for more steps...

Step 3.8.2.2.1

Apply the distributive property.

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

Step 3.8.2.2.2

Apply the distributive property.

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

Step 3.8.2.2.3

Apply the distributive property.

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

Step 3.8.2.3

Simplify and combine like terms.

Tap for more steps...

Step 3.8.2.3.1

Simplify each term.

Tap for more steps...

Step 3.8.2.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.8.2.3.1.2

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

Tap for more steps...

Step 3.8.2.3.1.2.1

Move $x^{2}$ .

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

Step 3.8.2.3.1.2.2

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

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

Step 3.8.2.3.1.2.3

Add $2$ and $2$ .

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

Step 3.8.2.3.1.3

Multiply $4$ by $4$ .

$\frac{8 x x'}{16 x^{4} + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.8.2.3.1.4

Multiply $- 3$ by $4$ .

$\frac{8 x x'}{16 x^{4} - 12 x^{2} - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.8.2.3.1.5

Multiply $4$ by $- 3$ .

$\frac{8 x x'}{16 x^{4} - 12 x^{2} - 12 x^{2} - 3 \cdot - 3 + 1}$

Step 3.8.2.3.1.6

Multiply $- 3$ by $- 3$ .

$\frac{8 x x'}{16 x^{4} - 12 x^{2} - 12 x^{2} + 9 + 1}$

Step 3.8.2.3.2

Subtract $12 x^{2}$ from $- 12 x^{2}$ .

$\frac{8 x x'}{16 x^{4} - 24 x^{2} + 9 + 1}$

Step 3.8.2.4

Add $9$ and $1$ .

$\frac{8 x x'}{16 x^{4} - 24 x^{2} + 10}$

Step 3.8.2.5

Factor $2$ out of $16 x^{4} - 24 x^{2} + 10$ .

Tap for more steps...

Step 3.8.2.5.1

Factor $2$ out of $16 x^{4}$ .

$\frac{8 x x'}{2 (8 x^{4}) - 24 x^{2} + 10}$

Step 3.8.2.5.2

Factor $2$ out of $- 24 x^{2}$ .

$\frac{8 x x'}{2 (8 x^{4}) + 2 (- 12 x^{2}) + 10}$

Step 3.8.2.5.3

Factor $2$ out of $10$ .

$\frac{8 x x'}{2 (8 x^{4}) + 2 (- 12 x^{2}) + 2 (5)}$

Step 3.8.2.5.4

Factor $2$ out of $2 (8 x^{4}) + 2 (- 12 x^{2})$ .

$\frac{8 x x'}{2 (8 x^{4} - 12 x^{2}) + 2 (5)}$

Step 3.8.2.5.5

Factor $2$ out of $2 (8 x^{4} - 12 x^{2}) + 2 (5)$ .

$\frac{8 x x'}{2 (8 x^{4} - 12 x^{2} + 5)}$

Step 3.8.3

Cancel the common factor of $8$ and $2$ .

Tap for more steps...

Step 3.8.3.1

Factor $2$ out of $8 x x'$ .

$\frac{2 (4 x x')}{2 (8 x^{4} - 12 x^{2} + 5)}$

Step 3.8.3.2

Cancel the common factors.

Tap for more steps...

Step 3.8.3.2.1

Cancel the common factor.

$\frac{2 (4 x x')}{2 (8 x^{4} - 12 x^{2} + 5)}$

Step 3.8.3.2.2

Rewrite the expression.

$\frac{4 x x'}{8 x^{4} - 12 x^{2} + 5}$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = \frac{4 x x'}{8 x^{4} - 12 x^{2} + 5}$

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $\frac{4 x x'}{8 x^{4} - 12 x^{2} + 5} = 1$ .

$\frac{4 x x'}{8 x^{4} - 12 x^{2} + 5} = 1$

Step 5.2

Multiply both sides by $8 x^{4} - 12 x^{2} + 5$ .

$\frac{4 x x'}{8 x^{4} - 12 x^{2} + 5} (8 x^{4} - 12 x^{2} + 5) = 1 (8 x^{4} - 12 x^{2} + 5)$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

Simplify the left side.

Tap for more steps...

Step 5.3.1.1

Cancel the common factor of $8 x^{4} - 12 x^{2} + 5$ .

Tap for more steps...

Step 5.3.1.1.1

Cancel the common factor.

$\frac{4 x x'}{8 x^{4} - 12 x^{2} + 5} (8 x^{4} - 12 x^{2} + 5) = 1 (8 x^{4} - 12 x^{2} + 5)$

Step 5.3.1.1.2

Rewrite the expression.

$4 x x' = 1 (8 x^{4} - 12 x^{2} + 5)$

Step 5.3.2

Simplify the right side.

Tap for more steps...

Step 5.3.2.1

Multiply $8 x^{4} - 12 x^{2} + 5$ by $1$ .

$4 x x' = 8 x^{4} - 12 x^{2} + 5$

Step 5.4

Divide each term in $4 x x' = 8 x^{4} - 12 x^{2} + 5$ by $4 x$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $4 x x' = 8 x^{4} - 12 x^{2} + 5$ by $4 x$ .

$\frac{4 x x'}{4 x} = \frac{8 x^{4}}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{4 x x'}{4 x} = \frac{8 x^{4}}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x x'}{x} = \frac{8 x^{4}}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.4.2.2.1

Cancel the common factor.

$\frac{x x'}{x} = \frac{8 x^{4}}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{8 x^{4}}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Simplify each term.

Tap for more steps...

Step 5.4.3.1.1

Cancel the common factor of $8$ and $4$ .

Tap for more steps...

Step 5.4.3.1.1.1

Factor $4$ out of $8 x^{4}$ .

$x' = \frac{4 (2 x^{4})}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.1.2.1

Factor $4$ out of $4 x$ .

$x' = \frac{4 (2 x^{4})}{4 (x)} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$x' = \frac{4 (2 x^{4})}{4 x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$x' = \frac{2 x^{4}}{x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.2

Cancel the common factor of $x^{4}$ and $x$ .

Tap for more steps...

Step 5.4.3.1.2.1

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

$x' = \frac{x (2 x^{3})}{x} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.2.2.1

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

$x' = \frac{x (2 x^{3})}{x^{1}} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.2.2.2

Factor $x$ out of $x^{1}$ .

$x' = \frac{x (2 x^{3})}{x \cdot 1} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.2.2.3

Cancel the common factor.

$x' = \frac{x (2 x^{3})}{x \cdot 1} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.2.2.4

Rewrite the expression.

$x' = \frac{2 x^{3}}{1} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.2.2.5

Divide $2 x^{3}$ by $1$ .

$x' = 2 x^{3} + \frac{- 12 x^{2}}{4 x} + \frac{5}{4 x}$

Step 5.4.3.1.3

Cancel the common factor of $- 12$ and $4$ .

Tap for more steps...

Step 5.4.3.1.3.1

Factor $4$ out of $- 12 x^{2}$ .

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

Step 5.4.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.3.2.1

Factor $4$ out of $4 x$ .

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

Step 5.4.3.1.3.2.2

Cancel the common factor.

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

Step 5.4.3.1.3.2.3

Rewrite the expression.

$x' = 2 x^{3} + \frac{- 3 x^{2}}{x} + \frac{5}{4 x}$

Step 5.4.3.1.4

Cancel the common factor of $x^{2}$ and $x$ .

Tap for more steps...

Step 5.4.3.1.4.1

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

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

Step 5.4.3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.4.2.1

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

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

Step 5.4.3.1.4.2.2

Factor $x$ out of $x^{1}$ .

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

Step 5.4.3.1.4.2.3

Cancel the common factor.

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

Step 5.4.3.1.4.2.4

Rewrite the expression.

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

Step 5.4.3.1.4.2.5

Divide $- 3 x$ by $1$ .

$x' = 2 x^{3} - 3 x + \frac{5}{4 x}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = 2 x^{3} - 3 x + \frac{5}{4 x}$