Calculus Examples

$y = \arcsin (\frac{1}{9 x + 6})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arcsin (\frac{1}{9 x + 6}))$

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 $g (x) = \frac{1}{9 x + 6}$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{1}{9 x + 6}$ .

$\frac{d}{d u_{1}} [\arcsin (u_{1})] \frac{d}{d x} [\frac{1}{9 x + 6}]$

Step 3.1.2

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

$\frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [\frac{1}{9 x + 6}]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\frac{1}{9 x + 6}$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}} \frac{d}{d x} [\frac{1}{9 x + 6}]$

Step 3.2

Rewrite $\frac{1}{9 x + 6}$ as ${(9 x + 6)}^{- 1}$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}} \frac{d}{d x} [{(9 x + 6)}^{- 1}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 9 x + 6$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $9 x + 6$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [9 x + 6])$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = - 1$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}} (- {u_{2}}^{- 2} \frac{d}{d x} [9 x + 6])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $9 x + 6$ .

$\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}} (- {(9 x + 6)}^{- 2} \frac{d}{d x} [9 x + 6])$

Step 3.4

Differentiate.

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Step 3.4.1

Combine $\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}}$ and ${(9 x + 6)}^{- 2}$ .

$- \frac{{(9 x + 6)}^{- 2}}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}}} \frac{d}{d x} [9 x + 6]$

Step 3.4.2

Move ${(9 x + 6)}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} \frac{d}{d x} [9 x + 6]$

Step 3.4.3

By the Sum Rule, the derivative of $9 x + 6$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [6]$ .

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} (\frac{d}{d x} [9 x] + \frac{d}{d x} [6])$

Step 3.4.4

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

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} (9 \frac{d}{d x} [x] + \frac{d}{d x} [6])$

Step 3.4.5

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

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} (9 \cdot 1 + \frac{d}{d x} [6])$

Step 3.4.6

Multiply $9$ by $1$ .

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} (9 + \frac{d}{d x} [6])$

Step 3.4.7

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} (9 + 0)$

Step 3.4.8

Combine fractions.

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Step 3.4.8.1

Add $9$ and $0$ .

$- \frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}} \cdot 9$

Step 3.4.8.2

Multiply $9$ by $- 1$ .

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

Step 3.4.8.3

Combine $- 9$ and $\frac{1}{\sqrt{1 - {(\frac{1}{9 x + 6})}^{2}} {(9 x + 6)}^{2}}$ .

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

Step 3.4.8.4

Move the negative in front of the fraction.

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the product rule to $\frac{1}{9 x + 6}$ .

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

Step 3.5.2

One to any power is one.

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

Step 4

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

$y' = - \frac{9}{\sqrt{1 - \frac{1}{{(9 x + 6)}^{2}}} {(9 x + 6)}^{2}}$

Step 5

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

$\frac{d y}{d x} = - \frac{9}{\sqrt{1 - \frac{1}{{(9 x + 6)}^{2}}} {(9 x + 6)}^{2}}$