Calculus Examples

$f (x) = x^{2} + 2 x$ , $x = 1$

Step 1

The percentage rate of change for the function is the value of the derivative (rate of change) at $1$ over the value of the function at $1$ .

$\frac{f' (1)}{f (1)}$

Step 2

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

$\frac{2 x + 2}{x^{2} + 2 x}$

Step 3

Factor $2$ out of $2 x + 2$ .

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Step 3.1

Factor $2$ out of $2 x$ .

$\frac{f' (x)}{f (x)} = \frac{2 (x) + 2}{x^{2} + 2 x}$

Step 3.2

Factor $2$ out of $2$ .

$\frac{f' (x)}{f (x)} = \frac{2 (x) + 2 (1)}{x^{2} + 2 x}$

Step 3.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$\frac{f' (x)}{f (x)} = \frac{2 (x + 1)}{x^{2} + 2 x}$

Step 4

Factor $x$ out of $x^{2} + 2 x$ .

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Step 4.1

Factor $x$ out of $x^{2}$ .

$\frac{f' (x)}{f (x)} = \frac{2 (x + 1)}{x \cdot x + 2 x}$

Step 4.2

Factor $x$ out of $2 x$ .

$\frac{f' (x)}{f (x)} = \frac{2 (x + 1)}{x \cdot x + x \cdot 2}$

Step 4.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$\frac{f' (x)}{f (x)} = \frac{2 (x + 1)}{x (x + 2)}$

Step 5

Evaluate the formula at $x = 1$ .

$\frac{2 ((1) + 1)}{(1) ((1) + 2)}$

Step 6

Simplify $\frac{2 ((1) + 1)}{(1) ((1) + 2)}$ .

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Step 6.1

Add $1$ and $1$ .

$\frac{2 \cdot 2}{1 (1 + 2)}$

Step 6.2

Multiply $1 + 2$ by $1$ .

$\frac{2 \cdot 2}{1 + 2}$

Step 6.3

Multiply $2$ by $2$ .

$\frac{4}{1 + 2}$

Step 6.4

Add $1$ and $2$ .

$\frac{4}{3}$

Step 7

Convert $\frac{4}{3}$ to a decimal.

$1.\overline{3}$

Step 8

Convert $1.\overline{3}$ to a percentage.

$133.\overline{3}$ %