Calculus Examples

$y = 4 \arcsin (\frac{x}{2})$

Step 1

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

$4 \frac{d}{d x} [\arcsin (\frac{x}{2})]$

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 $g (x) = \frac{x}{2}$ .

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Step 2.1

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

$4 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [\frac{x}{2}])$

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 3.3

Simplify terms.

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Step 3.3.1

Multiply $\frac{1}{2}$ by $\frac{4}{\sqrt{1 - {(\frac{x}{2})}^{2}}}$ .

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

Step 3.3.2

Cancel the common factor of $4$ and $2$ .

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Step 3.3.2.1

Factor $2$ out of $4$ .

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor $2$ out of $2 \sqrt{1 - {(\frac{x}{2})}^{2}}$ .

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

Step 3.3.2.2.2

Cancel the common factor.

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

Step 3.3.2.2.3

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

Multiply $\frac{2}{\sqrt{1 - {(\frac{x}{2})}^{2}}}$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{x}{2}$ .

$\frac{2}{\sqrt{1 - \frac{x^{2}}{2^{2}}}}$

Step 4.2

Raise $2$ to the power of $2$ .

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