Calculus Examples

$\frac{d}{d x} (\arcsin (x^{4}))$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [x^{4}]$

Step 1.2

The derivative of $\arcsin (u)$ with respect to $u$ is $\frac{1}{\sqrt{1 - u^{2}}}$ .

$\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [x^{4}]$

Step 1.3

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

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{\sqrt{1 - x^{4 \cdot 2}}} \frac{d}{d x} [x^{4}]$

Step 2.1.2

Multiply $4$ by $2$ .

$\frac{1}{\sqrt{1 - x^{8}}} \frac{d}{d x} [x^{4}]$

Step 2.2

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

$\frac{1}{\sqrt{1 - x^{8}}} (4 x^{3})$

Step 2.3

Combine fractions.

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Step 2.3.1

Combine $4$ and $\frac{1}{\sqrt{1 - x^{8}}}$ .

$\frac{4}{\sqrt{1 - x^{8}}} x^{3}$

Step 2.3.2

Combine $\frac{4}{\sqrt{1 - x^{8}}}$ and $x^{3}$ .

$\frac{4 x^{3}}{\sqrt{1 - x^{8}}}$