Calculus Examples

$f (x) = \ln (\frac{x + 9}{x - 9})$

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x + 9}{x - 9}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x + 9}{x - 9}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [\frac{x + 9}{x - 9}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x + 9}{x - 9}$ .

$\frac{1}{\frac{x + 9}{x - 9}} \frac{d}{d x} [\frac{x + 9}{x - 9}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{x + 9}{x - 9}$ .

$1 \frac{x - 9}{x + 9} \frac{d}{d x} [\frac{x + 9}{x - 9}]$

Step 3

Multiply $\frac{x - 9}{x + 9}$ by $1$ .

$\frac{x - 9}{x + 9} \frac{d}{d x} [\frac{x + 9}{x - 9}]$

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 $f (x) = x + 9$ and $g (x) = x - 9$ .

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

Step 5

Differentiate.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{x - 9}{x + 9} \cdot \frac{(x - 9) (1 + 0) - (x + 9) \frac{d}{d x} [x - 9]}{{(x - 9)}^{2}}$

Step 5.4

Simplify the expression.

Tap for more steps...

Step 5.4.1

Add $1$ and $0$ .

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

Step 5.4.2

Multiply $x - 9$ by $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

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

Step 5.8

Combine fractions.

Tap for more steps...

Step 5.8.1

Add $1$ and $0$ .

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

Step 5.8.2

Multiply $- 1$ by $1$ .

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

Step 5.8.3

Multiply $\frac{x - 9}{x + 9}$ by $\frac{x - 9 - (x + 9)}{{(x - 9)}^{2}}$ .

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

Step 6

Cancel the common factors.

Tap for more steps...

Step 6.1

Factor $x - 9$ out of $(x + 9) {(x - 9)}^{2}$ .

$\frac{(x - 9) (x - 9 - (x + 9))}{(x - 9) ((x + 9) (x - 9))}$

Step 6.2

Cancel the common factor.

$\frac{(x - 9) (x - 9 - (x + 9))}{(x - 9) ((x + 9) (x - 9))}$

Step 6.3

Rewrite the expression.

$\frac{x - 9 - (x + 9)}{(x + 9) (x - 9)}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

$\frac{x - 9 - x - 1 \cdot 9}{(x + 9) (x - 9)}$

Step 7.2

Simplify the numerator.

Tap for more steps...

Step 7.2.1

Combine the opposite terms in $x - 9 - x - 1 \cdot 9$ .

Tap for more steps...

Step 7.2.1.1

Subtract $x$ from $x$ .

$\frac{0 - 9 - 1 \cdot 9}{(x + 9) (x - 9)}$

Step 7.2.1.2

Subtract $9$ from $0$ .

$\frac{- 9 - 1 \cdot 9}{(x + 9) (x - 9)}$

Step 7.2.2

Multiply $- 1$ by $9$ .

$\frac{- 9 - 9}{(x + 9) (x - 9)}$

Step 7.2.3

Subtract $9$ from $- 9$ .

$\frac{- 18}{(x + 9) (x - 9)}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{18}{(x + 9) (x - 9)}$