Calculus Examples

$\ln (\frac{6 - x}{6 + x})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{6 - x}{6 + x}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [\frac{6 - x}{6 + x}]$

Step 1.3

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

$\frac{1}{\frac{6 - x}{6 + x}} \frac{d}{d x} [\frac{6 - x}{6 + x}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{6 - x}{6 + x}$ .

$1 \frac{6 + x}{6 - x} \frac{d}{d x} [\frac{6 - x}{6 + x}]$

Step 3

Multiply $\frac{6 + x}{6 - x}$ by $1$ .

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

Step 4

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 $f (x) = 6 - x$ and $g (x) = 6 + x$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 5.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 5.5

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

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

Step 5.6

Simplify the expression.

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Step 5.6.1

Multiply $- 1$ by $1$ .

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

Step 5.6.2

Move $- 1$ to the left of $6 + x$ .

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

Step 5.6.3

Rewrite $- 1 (6 + x)$ as $- (6 + x)$ .

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

Add $0$ and $\frac{d}{d x} [x]$ .

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

Step 5.10

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

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

Step 5.11

Combine fractions.

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Step 5.11.1

Multiply $- 1$ by $1$ .

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

Step 5.11.2

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

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

Step 6

Cancel the common factors.

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Step 6.1

Factor $6 + x$ out of $(6 - x) {(6 + x)}^{2}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Multiply $- 1$ by $6$ .

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

Step 7.3.1.2

Multiply $- 1$ by $6$ .

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

Step 7.3.1.3

Multiply $- - x$ .

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Step 7.3.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 6 - x - 6 + 1 x}{(6 - x) (6 + x)}$

Step 7.3.1.3.2

Multiply $x$ by $1$ .

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

Step 7.3.2

Combine the opposite terms in $- 6 - x - 6 + x$ .

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Step 7.3.2.1

Add $- x$ and $x$ .

$\frac{- 6 + 0 - 6}{(6 - x) (6 + x)}$

Step 7.3.2.2

Add $- 6$ and $0$ .

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

Step 7.3.3

Subtract $6$ from $- 6$ .

$\frac{- 12}{(6 - x) (6 + x)}$

Step 7.4

Move the negative in front of the fraction.

$- \frac{12}{(6 - x) (6 + x)}$