Find the Derivative – d/dx y=arcsin(3x)

$y = \arcsin (3 x)$

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

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

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

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

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

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

Differentiate.

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Factor $3$ out of $3 x$ .

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

Simplify the expression.

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

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

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

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

Multiply $9$ by $- 1$ .

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

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

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

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

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

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

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

Multiply $\frac{3}{\sqrt{1 - 9 x^{2}}}$ by $1$ .

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

Find the Derivative – d/dx y=arcsin(3x)

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