Calculus Examples

$\frac{8 + x}{8 - x}$

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 $f (x) = 8 + x$ and $g (x) = 8 - x$ .

$\frac{(8 - x) \frac{d}{d x} [8 + x] - (8 + x) \frac{d}{d x} [8 - x]}{{(8 - x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $8 + x$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [x]$ .

$\frac{(8 - x) (\frac{d}{d x} [8] + \frac{d}{d x} [x]) - (8 + x) \frac{d}{d x} [8 - x]}{{(8 - x)}^{2}}$

Step 2.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{(8 - x) (0 + \frac{d}{d x} [x]) - (8 + x) \frac{d}{d x} [8 - x]}{{(8 - x)}^{2}}$

Step 2.3

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

$\frac{(8 - x) \frac{d}{d x} [x] - (8 + x) \frac{d}{d x} [8 - x]}{{(8 - x)}^{2}}$

Step 2.4

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

$\frac{(8 - x) \cdot 1 - (8 + x) \frac{d}{d x} [8 - x]}{{(8 - x)}^{2}}$

Step 2.5

Multiply $8 - x$ by $1$ .

$\frac{8 - x - (8 + x) \frac{d}{d x} [8 - x]}{{(8 - x)}^{2}}$

Step 2.6

By the Sum Rule, the derivative of $8 - x$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x]$ .

$\frac{8 - x - (8 + x) (\frac{d}{d x} [8] + \frac{d}{d x} [- x])}{{(8 - x)}^{2}}$

Step 2.7

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{8 - x - (8 + x) (0 + \frac{d}{d x} [- x])}{{(8 - x)}^{2}}$

Step 2.8

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

$\frac{8 - x - (8 + x) \frac{d}{d x} [- x]}{{(8 - x)}^{2}}$

Step 2.9

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

$\frac{8 - x - (8 + x) (- \frac{d}{d x} [x])}{{(8 - x)}^{2}}$

Step 2.10

Multiply.

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

$\frac{8 - x + 1 (8 + x) \frac{d}{d x} [x]}{{(8 - x)}^{2}}$

Step 2.10.2

Multiply $8 + x$ by $1$ .

$\frac{8 - x + (8 + x) \frac{d}{d x} [x]}{{(8 - x)}^{2}}$

Step 2.11

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

$\frac{8 - x + (8 + x) \cdot 1}{{(8 - x)}^{2}}$

Step 2.12

Simplify by adding terms.

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Step 2.12.1

Multiply $8 + x$ by $1$ .

$\frac{8 - x + 8 + x}{{(8 - x)}^{2}}$

Step 2.12.2

Add $8$ and $8$ .

$\frac{- x + 16 + x}{{(8 - x)}^{2}}$

Step 2.12.3

Add $- x$ and $x$ .

$\frac{0 + 16}{{(8 - x)}^{2}}$

Step 2.12.4

Simplify the expression.

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Step 2.12.4.1

Add $0$ and $16$ .

$\frac{16}{{(8 - x)}^{2}}$

Step 2.12.4.2

Reorder terms.

$\frac{16}{{(- x + 8)}^{2}}$