Calculus Examples

$y = \arcsin (9 x^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arcsin (9 x^{3}))$

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

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 $g (x) = 9 x^{3}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $9 x^{3}$ .

$\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [9 x^{3}]$

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $9 x^{3}$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

Factor $9$ out of $9 x^{3}$ .

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Apply the product rule to $9 (x^{3})$ .

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

Step 3.2.2.2

Raise $9$ to the power of $2$ .

$\frac{1}{\sqrt{1 - (81 {(x^{3})}^{2})}} \frac{d}{d x} [9 x^{3}]$

Step 3.2.2.3

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

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Step 3.2.2.3.1

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

$\frac{1}{\sqrt{1 - (81 x^{3 \cdot 2})}} \frac{d}{d x} [9 x^{3}]$

Step 3.2.2.3.2

Multiply $3$ by $2$ .

$\frac{1}{\sqrt{1 - (81 x^{6})}} \frac{d}{d x} [9 x^{3}]$

Step 3.2.2.4

Multiply $81$ by $- 1$ .

$\frac{1}{\sqrt{1 - 81 x^{6}}} \frac{d}{d x} [9 x^{3}]$

Step 3.2.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{3}$ with respect to $x$ is $9 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{\sqrt{1 - 81 x^{6}}} (9 \frac{d}{d x} [x^{3}])$

Step 3.2.4

Combine $9$ and $\frac{1}{\sqrt{1 - 81 x^{6}}}$ .

$\frac{9}{\sqrt{1 - 81 x^{6}}} \frac{d}{d x} [x^{3}]$

Step 3.2.5

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

$\frac{9}{\sqrt{1 - 81 x^{6}}} (3 x^{2})$

Step 3.2.6

Combine fractions.

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Step 3.2.6.1

Combine $3$ and $\frac{9}{\sqrt{1 - 81 x^{6}}}$ .

$\frac{3 \cdot 9}{\sqrt{1 - 81 x^{6}}} x^{2}$

Step 3.2.6.2

Multiply $3$ by $9$ .

$\frac{27}{\sqrt{1 - 81 x^{6}}} x^{2}$

Step 3.2.6.3

Combine $\frac{27}{\sqrt{1 - 81 x^{6}}}$ and $x^{2}$ .

$\frac{27 x^{2}}{\sqrt{1 - 81 x^{6}}}$

Step 4

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

$y' = \frac{27 x^{2}}{\sqrt{1 - 81 x^{6}}}$

Step 5

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

$\frac{d y}{d x} = \frac{27 x^{2}}{\sqrt{1 - 81 x^{6}}}$