Calculus Examples

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

Step 1

Raise $16$ to the power of $4$ .

$\frac{d}{d x} [65536 \sqrt{4 x^{4} + 4}]$

Step 2

Rewrite $\frac{d}{d x} [65536 \sqrt{4 x^{4} + 4}]$ as $\frac{d}{d x} [65536 (2 \sqrt{x^{4} + 1})]$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$\frac{d}{d x} [65536 \sqrt{4 (x^{4}) + 4}]$

Step 2.1.2

Factor $4$ out of $4$ .

$\frac{d}{d x} [65536 \sqrt{4 (x^{4}) + 4 (1)}]$

Step 2.1.3

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

$\frac{d}{d x} [65536 \sqrt{4 (x^{4} + 1)}]$

Step 2.1.4

Rewrite $4$ as $2^{2}$ .

$\frac{d}{d x} [65536 \sqrt{2^{2} (x^{4} + 1)}]$

Step 2.2

Pull terms out from under the radical.

$\frac{d}{d x} [65536 (2 \sqrt{x^{4} + 1})]$

Step 3

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 3.1

Multiply $2$ by $65536$ .

$\frac{d}{d x} [131072 \sqrt{x^{4} + 1}]$

Step 3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} + 1}$ as ${(x^{4} + 1)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [131072 {(x^{4} + 1)}^{\frac{1}{2}}]$

Step 3.3

Since $131072$ is constant with respect to $x$ , the derivative of $131072 {(x^{4} + 1)}^{\frac{1}{2}}$ with respect to $x$ is $131072 \frac{d}{d x} [{(x^{4} + 1)}^{\frac{1}{2}}]$ .

$131072 \frac{d}{d x} [{(x^{4} + 1)}^{\frac{1}{2}}]$

Step 4

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

Tap for more steps...

Step 4.1

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

$131072 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{4} + 1])$

Step 4.2

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

$131072 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{4} + 1])$

Step 4.3

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

$131072 (\frac{1}{2} {(x^{4} + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{4} + 1])$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$131072 (\frac{1}{2} {(x^{4} + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{4} + 1])$

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

$131072 (\frac{1}{2} {(x^{4} + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{4} + 1])$

Step 7

Combine the numerators over the common denominator.

$131072 (\frac{1}{2} {(x^{4} + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{4} + 1])$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $2$ .

$131072 (\frac{1}{2} {(x^{4} + 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{4} + 1])$

Step 8.2

Subtract $2$ from $1$ .

$131072 (\frac{1}{2} {(x^{4} + 1)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{4} + 1])$

Step 9

Move the negative in front of the fraction.

$131072 (\frac{1}{2} {(x^{4} + 1)}^{- \frac{1}{2}} \frac{d}{d x} [x^{4} + 1])$

Step 10

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

$131072 (\frac{{(x^{4} + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{4} + 1])$

Step 11

Move ${(x^{4} + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$131072 (\frac{1}{2 {(x^{4} + 1)}^{\frac{1}{2}}} \frac{d}{d x} [x^{4} + 1])$

Step 12

Combine $\frac{1}{2 {(x^{4} + 1)}^{\frac{1}{2}}}$ and $131072$ .

$\frac{131072}{2 {(x^{4} + 1)}^{\frac{1}{2}}} \frac{d}{d x} [x^{4} + 1]$

Step 13

Factor $2$ out of $131072$ .

$\frac{2 \cdot 65536}{2 {(x^{4} + 1)}^{\frac{1}{2}}} \frac{d}{d x} [x^{4} + 1]$

Step 14

Cancel the common factors.

Tap for more steps...

Step 14.1

Factor $2$ out of $2 {(x^{4} + 1)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 65536}{2 ({(x^{4} + 1)}^{\frac{1}{2}})} \frac{d}{d x} [x^{4} + 1]$

Step 14.2

Cancel the common factor.

$\frac{2 \cdot 65536}{2 {(x^{4} + 1)}^{\frac{1}{2}}} \frac{d}{d x} [x^{4} + 1]$

Step 14.3

Rewrite the expression.

$\frac{65536}{{(x^{4} + 1)}^{\frac{1}{2}}} \frac{d}{d x} [x^{4} + 1]$

Step 15

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

$\frac{65536}{{(x^{4} + 1)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1])$

Step 16

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

$\frac{65536}{{(x^{4} + 1)}^{\frac{1}{2}}} (4 x^{3} + \frac{d}{d x} [1])$

Step 17

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

$\frac{65536}{{(x^{4} + 1)}^{\frac{1}{2}}} (4 x^{3} + 0)$

Step 18

Combine fractions.

Tap for more steps...

Step 18.1

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

$\frac{65536}{{(x^{4} + 1)}^{\frac{1}{2}}} (4 x^{3})$

Step 18.2

Combine $4$ and $\frac{65536}{{(x^{4} + 1)}^{\frac{1}{2}}}$ .

$\frac{4 \cdot 65536}{{(x^{4} + 1)}^{\frac{1}{2}}} x^{3}$

Step 18.3

Multiply $4$ by $65536$ .

$\frac{262144}{{(x^{4} + 1)}^{\frac{1}{2}}} x^{3}$

Step 18.4

Combine $\frac{262144}{{(x^{4} + 1)}^{\frac{1}{2}}}$ and $x^{3}$ .

$\frac{262144 x^{3}}{{(x^{4} + 1)}^{\frac{1}{2}}}$