Calculus Examples

$f (x) = \frac{x}{8 x - 8}$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Multiply $8 x - 8$ by $1$ .

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

Step 2.3

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

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

Step 2.4

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

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

Step 2.5

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

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

Step 2.6

Multiply $8$ by $1$ .

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

Step 2.7

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

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

Step 2.8

Simplify by adding terms.

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Step 2.8.1

Add $8$ and $0$ .

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

Step 2.8.2

Multiply $8$ by $- 1$ .

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

Step 2.8.3

Subtract $8 x$ from $8 x$ .

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

Step 2.8.4

Simplify the expression.

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Step 2.8.4.1

Subtract $8$ from $0$ .

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

Step 2.8.4.2

Move the negative in front of the fraction.

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Factor $8$ out of $8 x - 8$ .

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Step 3.1.1.1

Factor $8$ out of $8 x$ .

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

Step 3.1.1.2

Factor $8$ out of $- 8$ .

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

Step 3.1.1.3

Factor $8$ out of $8 (x) + 8 (- 1)$ .

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

Step 3.1.2

Apply the product rule to $8 (x - 1)$ .

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

Step 3.1.3

Raise $8$ to the power of $2$ .

$- \frac{8}{64 {(x - 1)}^{2}}$

Step 3.2

Cancel the common factor of $8$ and $64$ .

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Step 3.2.1

Factor $8$ out of $8$ .

$- \frac{8 (1)}{64 {(x - 1)}^{2}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $8$ out of $64 {(x - 1)}^{2}$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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