Calculus Examples

$y = 4 \arcsin (2 x^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (4 \arcsin (2 x^{3}))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \arcsin (2 x^{3})$ with respect to $x$ is $4 \frac{d}{d x} [\arcsin (2 x^{3})]$ .

$4 \frac{d}{d x} [\arcsin (2 x^{3})]$

Step 3.2

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 $g (x) = 2 x^{3}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $2 x^{3}$ .

$4 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [2 x^{3}])$

Step 3.2.2

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

$4 (\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [2 x^{3}])$

Step 3.2.3

Replace all occurrences of $u$ with $2 x^{3}$ .

$4 (\frac{1}{\sqrt{1 - {(2 x^{3})}^{2}}} \frac{d}{d x} [2 x^{3}])$

Step 3.3

Differentiate.

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Step 3.3.1

Factor $2$ out of $2 x^{3}$ .

$4 (\frac{1}{\sqrt{1 - {(2 (x^{3}))}^{2}}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2

Combine fractions.

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Step 3.3.2.1

Simplify the expression.

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Step 3.3.2.1.1

Apply the product rule to $2 (x^{3})$ .

$4 (\frac{1}{\sqrt{1 - (2^{2} {(x^{3})}^{2})}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2.1.2

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

$4 (\frac{1}{\sqrt{1 - (4 {(x^{3})}^{2})}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2.1.3

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

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Step 3.3.2.1.3.1

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

$4 (\frac{1}{\sqrt{1 - (4 x^{3 \cdot 2})}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2.1.3.2

Multiply $3$ by $2$ .

$4 (\frac{1}{\sqrt{1 - (4 x^{6})}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2.1.4

Multiply $4$ by $- 1$ .

$4 (\frac{1}{\sqrt{1 - 4 x^{6}}} \frac{d}{d x} [2 x^{3}])$

Step 3.3.2.2

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

$\frac{4}{\sqrt{1 - 4 x^{6}}} \frac{d}{d x} [2 x^{3}]$

Step 3.3.3

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

$\frac{4}{\sqrt{1 - 4 x^{6}}} (2 \frac{d}{d x} [x^{3}])$

Step 3.3.4

Combine fractions.

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Step 3.3.4.1

Combine $2$ and $\frac{4}{\sqrt{1 - 4 x^{6}}}$ .

$\frac{2 \cdot 4}{\sqrt{1 - 4 x^{6}}} \frac{d}{d x} [x^{3}]$

Step 3.3.4.2

Multiply $2$ by $4$ .

$\frac{8}{\sqrt{1 - 4 x^{6}}} \frac{d}{d x} [x^{3}]$

Step 3.3.5

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

$\frac{8}{\sqrt{1 - 4 x^{6}}} (3 x^{2})$

Step 3.3.6

Combine fractions.

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Step 3.3.6.1

Combine $3$ and $\frac{8}{\sqrt{1 - 4 x^{6}}}$ .

$\frac{3 \cdot 8}{\sqrt{1 - 4 x^{6}}} x^{2}$

Step 3.3.6.2

Multiply $3$ by $8$ .

$\frac{24}{\sqrt{1 - 4 x^{6}}} x^{2}$

Step 3.3.6.3

Combine $\frac{24}{\sqrt{1 - 4 x^{6}}}$ and $x^{2}$ .

$\frac{24 x^{2}}{\sqrt{1 - 4 x^{6}}}$

Step 4

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

$y' = \frac{24 x^{2}}{\sqrt{1 - 4 x^{6}}}$

Step 5

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

$\frac{d y}{d x} = \frac{24 x^{2}}{\sqrt{1 - 4 x^{6}}}$