Calculus Examples

${(\frac{x + 6}{x - 6})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \frac{x + 6}{x - 6}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x + 6}{x - 6}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [\frac{x + 6}{x - 6}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [\frac{x + 6}{x - 6}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x + 6}{x - 6}$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 6$ and $g (x) = x - 6$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$3 {(\frac{x + 6}{x - 6})}^{2} \frac{(x - 6) (1 + 0) - (x + 6) \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $x - 6$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

$3 {(\frac{x + 6}{x - 6})}^{2} \frac{x - 6 - (x + 6) (1 + 0)}{{(x - 6)}^{2}}$

Step 3.8

Combine fractions.

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Step 3.8.1

Add $1$ and $0$ .

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

Step 3.8.2

Multiply $- 1$ by $1$ .

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

Step 3.8.3

Combine $\frac{x - 6 - (x + 6)}{{(x - 6)}^{2}}$ and $3$ .

$\frac{(x - 6 - (x + 6)) \cdot 3}{{(x - 6)}^{2}} {(\frac{x + 6}{x - 6})}^{2}$

Step 3.8.4

Move $3$ to the left of $x - 6 - (x + 6)$ .

$\frac{3 \cdot (x - 6 - (x + 6))}{{(x - 6)}^{2}} {(\frac{x + 6}{x - 6})}^{2}$

Step 4

Simplify.

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Step 4.1

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

$\frac{3 (x - 6 - (x + 6))}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $3$ by $- 6$ .

$\frac{3 x - 18 + 3 (- x) + 3 (- 1 \cdot 6)}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.2

Multiply $- 1$ by $3$ .

$\frac{3 x - 18 - 3 x + 3 (- 1 \cdot 6)}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.3

Multiply $- 1$ by $6$ .

$\frac{3 x - 18 - 3 x + 3 \cdot - 6}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.4

Multiply $3$ by $- 6$ .

$\frac{3 x - 18 - 3 x - 18}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.5

Subtract $3 x$ from $3 x$ .

$\frac{0 - 18 - 18}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.6

Subtract $18$ from $0$ .

$\frac{- 18 - 18}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.7

Subtract $18$ from $- 18$ .

$\frac{- 36}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.8

Move the negative in front of the fraction.

$- \frac{36}{{(x - 6)}^{2}} \cdot \frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$

Step 4.4.9

Multiply $\frac{{(x + 6)}^{2}}{{(x - 6)}^{2}}$ by $\frac{36}{{(x - 6)}^{2}}$ .

$- \frac{{(x + 6)}^{2} \cdot 36}{{(x - 6)}^{2} {(x - 6)}^{2}}$

Step 4.4.10

Multiply ${(x - 6)}^{2}$ by ${(x - 6)}^{2}$ by adding the exponents.

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Step 4.4.10.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{{(x + 6)}^{2} \cdot 36}{{(x - 6)}^{2 + 2}}$

Step 4.4.10.2

Add $2$ and $2$ .

$- \frac{{(x + 6)}^{2} \cdot 36}{{(x - 6)}^{4}}$

Step 4.4.11

Move $36$ to the left of ${(x + 6)}^{2}$ .

$- \frac{36 {(x + 6)}^{2}}{{(x - 6)}^{4}}$