Precalculus Examples

$f (x) = \frac{9 x}{2 x - 13}$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9 x}{2 x - 13}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{x}{2 x - 13}]$ .

$9 \frac{d}{d x} [\frac{x}{2 x - 13}]$

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 $f (x) = x$ and $g (x) = 2 x - 13$ .

$9 \frac{(2 x - 13) \frac{d}{d x} [x] - x \frac{d}{d x} [2 x - 13]}{{(2 x - 13)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$9 \frac{(2 x - 13) \cdot 1 - x \frac{d}{d x} [2 x - 13]}{{(2 x - 13)}^{2}}$

Step 3.2

Multiply $2 x - 13$ by $1$ .

$9 \frac{2 x - 13 - x \frac{d}{d x} [2 x - 13]}{{(2 x - 13)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $2 x - 13$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 13]$ .

$9 \frac{2 x - 13 - x (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 13])}{{(2 x - 13)}^{2}}$

Step 3.4

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$9 \frac{2 x - 13 - x (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 13])}{{(2 x - 13)}^{2}}$

Step 3.5

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

$9 \frac{2 x - 13 - x (2 \cdot 1 + \frac{d}{d x} [- 13])}{{(2 x - 13)}^{2}}$

Step 3.6

Multiply $2$ by $1$ .

$9 \frac{2 x - 13 - x (2 + \frac{d}{d x} [- 13])}{{(2 x - 13)}^{2}}$

Step 3.7

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13$ with respect to $x$ is $0$ .

$9 \frac{2 x - 13 - x (2 + 0)}{{(2 x - 13)}^{2}}$

Step 3.8

Simplify terms.

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Step 3.8.1

Add $2$ and $0$ .

$9 \frac{2 x - 13 - x \cdot 2}{{(2 x - 13)}^{2}}$

Step 3.8.2

Multiply $2$ by $- 1$ .

$9 \frac{2 x - 13 - 2 x}{{(2 x - 13)}^{2}}$

Step 3.8.3

Subtract $2 x$ from $2 x$ .

$9 \frac{0 - 13}{{(2 x - 13)}^{2}}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Subtract $13$ from $0$ .

$9 \frac{- 13}{{(2 x - 13)}^{2}}$

Step 3.8.4.2

Move the negative in front of the fraction.

$9 (- \frac{13}{{(2 x - 13)}^{2}})$

Step 3.8.4.3

Multiply $- 1$ by $9$ .

$- 9 \frac{13}{{(2 x - 13)}^{2}}$

Step 3.8.5

Combine $- 9$ and $\frac{13}{{(2 x - 13)}^{2}}$ .

$\frac{- 9 \cdot 13}{{(2 x - 13)}^{2}}$

Step 3.8.6

Simplify the expression.

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Step 3.8.6.1

Multiply $- 9$ by $13$ .

$\frac{- 117}{{(2 x - 13)}^{2}}$

Step 3.8.6.2

Move the negative in front of the fraction.

$- \frac{117}{{(2 x - 13)}^{2}}$