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

$y = \arcsin (x^{3})$

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 $g (x) = x^{3}$ .

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

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

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

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

Replace all occurrences of $u$ with $x^{3}$ .

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

Differentiate using the Power Rule.

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Multiply the exponents in ${(x^{3})}^{2}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $3$ by $2$ .

$\frac{1}{\sqrt{1 - x^{6}}} \frac{d}{d x} [x^{3}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{\sqrt{1 - x^{6}}} (3 x^{2})$

Combine fractions.

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Combine $3$ and $\frac{1}{\sqrt{1 - x^{6}}}$ .

$\frac{3}{\sqrt{1 - x^{6}}} x^{2}$

Combine $\frac{3}{\sqrt{1 - x^{6}}}$ and $x^{2}$ .

$\frac{3 x^{2}}{\sqrt{1 - x^{6}}}$

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

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