Calculus Examples

$2 \arcsin (5 x)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \arcsin (5 x)$ with respect to $x$ is $2 \frac{d}{d x} [\arcsin (5 x)]$ .

$2 \frac{d}{d x} [\arcsin (5 x)]$

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 $g (x) = 5 x$ .

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To apply the Chain Rule, set $u$ as $5 x$ .

$2 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [5 x])$

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

$2 (\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [5 x])$

Replace all occurrences of $u$ with $5 x$ .

$2 (\frac{1}{\sqrt{1 - {(5 x)}^{2}}} \frac{d}{d x} [5 x])$

Step 3

Differentiate.

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Factor $5$ out of $5 x$ .

$2 (\frac{1}{\sqrt{1 - {(5 (x))}^{2}}} \frac{d}{d x} [5 x])$

Combine fractions.

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Simplify the expression.

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Apply the product rule to $5 (x)$ .

$2 (\frac{1}{\sqrt{1 - (5^{2} x^{2})}} \frac{d}{d x} [5 x])$

Raise $5$ to the power of $2$ .

$2 (\frac{1}{\sqrt{1 - (25 x^{2})}} \frac{d}{d x} [5 x])$

Multiply $25$ by $- 1$ .

$2 (\frac{1}{\sqrt{1 - 25 x^{2}}} \frac{d}{d x} [5 x])$

Combine $\frac{1}{\sqrt{1 - 25 x^{2}}}$ and $2$ .

$\frac{2}{\sqrt{1 - 25 x^{2}}} \frac{d}{d x} [5 x]$

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{2}{\sqrt{1 - 25 x^{2}}} (5 \frac{d}{d x} [x])$

Combine fractions.

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Combine $5$ and $\frac{2}{\sqrt{1 - 25 x^{2}}}$ .

$\frac{5 \cdot 2}{\sqrt{1 - 25 x^{2}}} \frac{d}{d x} [x]$

Multiply $5$ by $2$ .

$\frac{10}{\sqrt{1 - 25 x^{2}}} \frac{d}{d x} [x]$

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

$\frac{10}{\sqrt{1 - 25 x^{2}}} \cdot 1$

Multiply $\frac{10}{\sqrt{1 - 25 x^{2}}}$ by $1$ .

$\frac{10}{\sqrt{1 - 25 x^{2}}}$