Calculus Examples

${(\arcsin (x))}^{2}$

Step 1

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

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To apply the Chain Rule, set $u$ as $\arcsin (x)$ .

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

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

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

Replace all occurrences of $u$ with $\arcsin (x)$ .

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

Step 2

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

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

Step 3

Combine fractions.

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

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

Combine $\frac{2}{\sqrt{1 - x^{2}}}$ and $\arcsin (x)$ .

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