Calculus Examples

$g (x) = \frac{2 x - 4}{7 x + 8}$ , $x = 8$

Step 1

The percentage rate of change for the function is the value of the derivative (rate of change) at $8$ over the value of the function at $8$ .

$\frac{f' (8)}{f (8)}$

Step 2

Substitute the functions into the formula to find the function for the percentage rate of change.

$\frac{\frac{44}{{(7 x + 8)}^{2}}}{\frac{2 x - 4}{7 x + 8}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{f' (x)}{f (x)} = \frac{44}{{(7 x + 8)}^{2}} \cdot \frac{7 x + 8}{2 x - 4}$

Step 4

Cancel the common factor of $7 x + 8$ .

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Step 4.1

Factor $7 x + 8$ out of ${(7 x + 8)}^{2}$ .

$\frac{f' (x)}{f (x)} = \frac{44}{(7 x + 8) (7 x + 8)} \cdot \frac{7 x + 8}{2 x - 4}$

Step 4.2

Cancel the common factor.

$\frac{f' (x)}{f (x)} = \frac{44}{(7 x + 8) (7 x + 8)} \cdot \frac{7 x + 8}{2 x - 4}$

Step 4.3

Rewrite the expression.

$\frac{f' (x)}{f (x)} = \frac{44}{7 x + 8} \cdot \frac{1}{2 x - 4}$

Step 5

Multiply $\frac{44}{7 x + 8}$ by $\frac{1}{2 x - 4}$ .

$\frac{f' (x)}{f (x)} = \frac{44}{(7 x + 8) (2 x - 4)}$

Step 6

Cancel the common factor of $44$ and $2 x - 4$ .

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Step 6.1

Factor $2$ out of $44$ .

$\frac{f' (x)}{f (x)} = \frac{2 \cdot 22}{(7 x + 8) (2 x - 4)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $(7 x + 8) (2 x - 4)$ .

$\frac{f' (x)}{f (x)} = \frac{2 \cdot 22}{2 ((7 x + 8) (x - 2))}$

Step 6.2.2

Cancel the common factor.

$\frac{f' (x)}{f (x)} = \frac{2 \cdot 22}{2 ((7 x + 8) (x - 2))}$

Step 6.2.3

Rewrite the expression.

$\frac{f' (x)}{f (x)} = \frac{22}{(7 x + 8) (x - 2)}$

Step 7

Evaluate the formula at $x = 8$ .

$\frac{22}{(7 (8) + 8) ((8) - 2)}$

Step 8

Simplify $\frac{22}{(7 (8) + 8) ((8) - 2)}$ .

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Step 8.1

Cancel the common factor of $22$ and $7 (8) + 8$ .

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Step 8.1.1

Factor $2$ out of $22$ .

$\frac{2 (11)}{(7 (8) + 8) ((8) - 2)}$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $2$ out of $(7 (8) + 8) ((8) - 2)$ .

$\frac{2 (11)}{2 ((7 \cdot (4) + 4) ((8) - 2))}$

Step 8.1.2.2

Cancel the common factor.

$\frac{2 \cdot 11}{2 ((7 \cdot (4) + 4) ((8) - 2))}$

Step 8.1.2.3

Rewrite the expression.

$\frac{11}{(7 \cdot (4) + 4) ((8) - 2)}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $7$ by $4$ .

$\frac{11}{(28 + 4) (8 - 2)}$

Step 8.2.2

Add $28$ and $4$ .

$\frac{11}{32 (8 - 2)}$

Step 8.2.3

Subtract $2$ from $8$ .

$\frac{11}{32 \cdot 6}$

Step 8.3

Multiply $32$ by $6$ .

$\frac{11}{192}$

Step 9

Convert $\frac{11}{192}$ to a decimal.

$0.057291\overline{6}$

Step 10

Convert $0.057291\overline{6}$ to a percentage.

$5.7291\overline{6}$ %