Calculus Examples

$y = 2 \arcsin (3 x)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \arcsin (3 x)$ with respect to $x$ is $2 \frac{d}{d x} [\arcsin (3 x)]$ .

$2 \frac{d}{d x} [\arcsin (3 x)]$

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 $g (x) = 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$2 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [3 x])$

Step 2.2

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

$2 (\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [3 x])$

Step 2.3

Replace all occurrences of $u$ with $3 x$ .

$2 (\frac{1}{\sqrt{1 - {(3 x)}^{2}}} \frac{d}{d x} [3 x])$

Step 3

Differentiate.

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Step 3.1

Factor $3$ out of $3 x$ .

$2 (\frac{1}{\sqrt{1 - {(3 (x))}^{2}}} \frac{d}{d x} [3 x])$

Step 3.2

Combine fractions.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Apply the product rule to $3 (x)$ .

$2 (\frac{1}{\sqrt{1 - (3^{2} x^{2})}} \frac{d}{d x} [3 x])$

Step 3.2.1.2

Raise $3$ to the power of $2$ .

$2 (\frac{1}{\sqrt{1 - (9 x^{2})}} \frac{d}{d x} [3 x])$

Step 3.2.1.3

Multiply $9$ by $- 1$ .

$2 (\frac{1}{\sqrt{1 - 9 x^{2}}} \frac{d}{d x} [3 x])$

Step 3.2.2

Combine $\frac{1}{\sqrt{1 - 9 x^{2}}}$ and $2$ .

$\frac{2}{\sqrt{1 - 9 x^{2}}} \frac{d}{d x} [3 x]$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{2}{\sqrt{1 - 9 x^{2}}} (3 \frac{d}{d x} [x])$

Step 3.4

Combine fractions.

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Step 3.4.1

Combine $3$ and $\frac{2}{\sqrt{1 - 9 x^{2}}}$ .

$\frac{3 \cdot 2}{\sqrt{1 - 9 x^{2}}} \frac{d}{d x} [x]$

Step 3.4.2

Multiply $3$ by $2$ .

$\frac{6}{\sqrt{1 - 9 x^{2}}} \frac{d}{d x} [x]$

Step 3.5

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

$\frac{6}{\sqrt{1 - 9 x^{2}}} \cdot 1$

Step 3.6

Multiply $\frac{6}{\sqrt{1 - 9 x^{2}}}$ by $1$ .

$\frac{6}{\sqrt{1 - 9 x^{2}}}$