Calculus Examples

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

Step 1

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 $f (x) = 9 + x$ and $g (x) = 1 - 9 x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $1 - 9 x$ by $1$ .

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

Step 2.6

By the Sum Rule, the derivative of $1 - 9 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 9 x]$ .

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

Step 2.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.8

Add $0$ and $\frac{d}{d x} [- 9 x]$ .

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

Step 2.9

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

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

Step 2.10

Multiply $- 9$ by $- 1$ .

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

Step 2.11

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

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

Step 2.12

Multiply $9$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

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Step 3.2.1.1

Add $- 9 x$ and $9 x$ .

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

Step 3.2.1.2

Add $1 + 9 \cdot 9$ and $0$ .

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

Step 3.2.2

Multiply $9$ by $9$ .

$\frac{1 + 81}{{(1 - 9 x)}^{2}}$

Step 3.2.3

Add $1$ and $81$ .

$\frac{82}{{(1 - 9 x)}^{2}}$

Step 3.3

Reorder terms.

$\frac{82}{{(- 9 x + 1)}^{2}}$