Calculus Examples

$f (x) = 8 \arcsin (x^{2})$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $8 \arcsin (x^{2})$ with respect to $x$ is $8 \frac{d}{d x} [\arcsin (x^{2})]$ .

$8 \frac{d}{d x} [\arcsin (x^{2})]$

Step 2

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

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Step 2.1

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

$8 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [x^{2}])$

Step 2.2

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

$8 (\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [x^{2}])$

Step 2.3

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

$8 (\frac{1}{\sqrt{1 - {(x^{2})}^{2}}} \frac{d}{d x} [x^{2}])$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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Step 3.1.1

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

$8 (\frac{1}{\sqrt{1 - x^{2 \cdot 2}}} \frac{d}{d x} [x^{2}])$

Step 3.1.2

Multiply $2$ by $2$ .

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

Step 3.2

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

$\frac{8}{\sqrt{1 - x^{4}}} \frac{d}{d x} [x^{2}]$

Step 3.3

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

$\frac{8}{\sqrt{1 - x^{4}}} (2 x)$

Step 3.4

Combine fractions.

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Step 3.4.1

Combine $2$ and $\frac{8}{\sqrt{1 - x^{4}}}$ .

$\frac{2 \cdot 8}{\sqrt{1 - x^{4}}} x$

Step 3.4.2

Multiply $2$ by $8$ .

$\frac{16}{\sqrt{1 - x^{4}}} x$

Step 3.4.3

Combine $\frac{16}{\sqrt{1 - x^{4}}}$ and $x$ .

$\frac{16 x}{\sqrt{1 - x^{4}}}$