Calculus Examples

$y = 16 \sqrt[4]{4 x^{4} + 4}$

Step 1

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

$y = 16 {(4 x^{4} + 4)}^{\frac{1}{4}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (16 {(4 x^{4} + 4)}^{\frac{1}{4}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Since $16$ is constant with respect to $x$ , the derivative of $16 {(4 x^{4} + 4)}^{\frac{1}{4}}$ with respect to $x$ is $16 \frac{d}{d x} [{(4 x^{4} + 4)}^{\frac{1}{4}}]$ .

$16 \frac{d}{d x} [{(4 x^{4} + 4)}^{\frac{1}{4}}]$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

$16 (\frac{d}{d u} [u^{\frac{1}{4}}] \frac{d}{d x} [4 x^{4} + 4])$

Step 4.2.2

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

$16 (\frac{1}{4} u^{\frac{1}{4} - 1} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.2.3

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

$16 (\frac{1}{4} {(4 x^{4} + 4)}^{\frac{1}{4} - 1} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$16 (\frac{1}{4} {(4 x^{4} + 4)}^{\frac{1}{4} - 1 \cdot \frac{4}{4}} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.4

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

$16 (\frac{1}{4} {(4 x^{4} + 4)}^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.5

Combine the numerators over the common denominator.

$16 (\frac{1}{4} {(4 x^{4} + 4)}^{\frac{1 - 1 \cdot 4}{4}} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 1$ by $4$ .

$16 (\frac{1}{4} {(4 x^{4} + 4)}^{\frac{1 - 4}{4}} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.6.2

Subtract $4$ from $1$ .

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

Step 4.7

Move the negative in front of the fraction.

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

Step 4.8

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

$16 (\frac{{(4 x^{4} + 4)}^{- \frac{3}{4}}}{4} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.9

Move ${(4 x^{4} + 4)}^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$16 (\frac{1}{4 {(4 x^{4} + 4)}^{\frac{3}{4}}} \frac{d}{d x} [4 x^{4} + 4])$

Step 4.10

Combine $\frac{1}{4 {(4 x^{4} + 4)}^{\frac{3}{4}}}$ and $16$ .

$\frac{16}{4 {(4 x^{4} + 4)}^{\frac{3}{4}}} \frac{d}{d x} [4 x^{4} + 4]$

Step 4.11

Factor $4$ out of $16$ .

$\frac{4 \cdot 4}{4 {(4 x^{4} + 4)}^{\frac{3}{4}}} \frac{d}{d x} [4 x^{4} + 4]$

Step 4.12

Cancel the common factors.

Tap for more steps...

Step 4.12.1

Factor $4$ out of $4 {(4 x^{4} + 4)}^{\frac{3}{4}}$ .

$\frac{4 \cdot 4}{4 ({(4 x^{4} + 4)}^{\frac{3}{4}})} \frac{d}{d x} [4 x^{4} + 4]$

Step 4.12.2

Cancel the common factor.

$\frac{4 \cdot 4}{4 {(4 x^{4} + 4)}^{\frac{3}{4}}} \frac{d}{d x} [4 x^{4} + 4]$

Step 4.12.3

Rewrite the expression.

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} \frac{d}{d x} [4 x^{4} + 4]$

Step 4.13

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

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} (\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [4])$

Step 4.14

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

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} (4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [4])$

Step 4.15

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

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} (4 (4 x^{3}) + \frac{d}{d x} [4])$

Step 4.16

Multiply $4$ by $4$ .

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} (16 x^{3} + \frac{d}{d x} [4])$

Step 4.17

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} (16 x^{3} + 0)$

Step 4.18

Combine fractions.

Tap for more steps...

Step 4.18.1

Add $16 x^{3}$ and $0$ .

$\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} (16 x^{3})$

Step 4.18.2

Combine $16$ and $\frac{4}{{(4 x^{4} + 4)}^{\frac{3}{4}}}$ .

$\frac{16 \cdot 4}{{(4 x^{4} + 4)}^{\frac{3}{4}}} x^{3}$

Step 4.18.3

Multiply $16$ by $4$ .

$\frac{64}{{(4 x^{4} + 4)}^{\frac{3}{4}}} x^{3}$

Step 4.18.4

Combine $\frac{64}{{(4 x^{4} + 4)}^{\frac{3}{4}}}$ and $x^{3}$ .

$\frac{64 x^{3}}{{(4 x^{4} + 4)}^{\frac{3}{4}}}$

Step 5

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

$y' = \frac{64 x^{3}}{{(4 x^{4} + 4)}^{\frac{3}{4}}}$

Step 6

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

$\frac{d y}{d x} = \frac{64 x^{3}}{{(4 x^{4} + 4)}^{\frac{3}{4}}}$