Find dy/dx y=arcsin(6x^4)

$y = \arcsin (6 x^{4})$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arcsin (6 x^{4}))$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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 $g (x) = 6 x^{4}$ .

To apply the Chain Rule, set $u$ as $6 x^{4}$ .

$\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [6 x^{4}]$

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

$\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [6 x^{4}]$

Replace all occurrences of $u$ with $6 x^{4}$ .

$\frac{1}{\sqrt{1 - {(6 x^{4})}^{2}}} \frac{d}{d x} [6 x^{4}]$

Differentiate.

Factor $6$ out of $6 x^{4}$ .

$\frac{1}{\sqrt{1 - {(6 (x^{4}))}^{2}}} \frac{d}{d x} [6 x^{4}]$

Simplify the expression.

Apply the product rule to $6 (x^{4})$ .

$\frac{1}{\sqrt{1 - (6^{2} {(x^{4})}^{2})}} \frac{d}{d x} [6 x^{4}]$

Raise $6$ to the power of $2$ .

$\frac{1}{\sqrt{1 - (36 {(x^{4})}^{2})}} \frac{d}{d x} [6 x^{4}]$

Multiply the exponents in ${(x^{4})}^{2}$ .

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

$\frac{1}{\sqrt{1 - (36 x^{4 \cdot 2})}} \frac{d}{d x} [6 x^{4}]$

Multiply $4$ by $2$ .

$\frac{1}{\sqrt{1 - (36 x^{8})}} \frac{d}{d x} [6 x^{4}]$

Multiply $36$ by $- 1$ .

$\frac{1}{\sqrt{1 - 36 x^{8}}} \frac{d}{d x} [6 x^{4}]$

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$\frac{1}{\sqrt{1 - 36 x^{8}}} (6 \frac{d}{d x} [x^{4}])$

Combine $6$ and $\frac{1}{\sqrt{1 - 36 x^{8}}}$ .

$\frac{6}{\sqrt{1 - 36 x^{8}}} \frac{d}{d x} [x^{4}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{6}{\sqrt{1 - 36 x^{8}}} (4 x^{3})$

Combine fractions.

Combine $4$ and $\frac{6}{\sqrt{1 - 36 x^{8}}}$ .

$\frac{4 \cdot 6}{\sqrt{1 - 36 x^{8}}} x^{3}$

Multiply $4$ by $6$ .

$\frac{24}{\sqrt{1 - 36 x^{8}}} x^{3}$

Combine $\frac{24}{\sqrt{1 - 36 x^{8}}}$ and $x^{3}$ .

$\frac{24 x^{3}}{\sqrt{1 - 36 x^{8}}}$

Reform the equation by setting the left side equal to the right side.

$y' = \frac{24 x^{3}}{\sqrt{1 - 36 x^{8}}}$

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{24 x^{3}}{\sqrt{1 - 36 x^{8}}}$

Find dy/dx y=arcsin(6x^4)

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