Calculus Examples

$f (x) = \sqrt{x^{2} + 9}$ , $x = 4$

Step 1

The percentage rate of change for the function is the value of the derivative (rate of change) at $4$ over the value of the function at $4$ .

$\frac{f' (4)}{f (4)}$

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{x^{2} + 9}}$

Step 4

Combine fractions.

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Step 4.1

Combine.

$\frac{f' (x)}{f (x)} = \frac{x \cdot 1}{{(x^{2} + 9)}^{\frac{1}{2}} \sqrt{x^{2} + 9}}$

Step 4.2

Multiply $x$ by $1$ .

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{1}{2}} \sqrt{x^{2} + 9}}$

Step 5

Simplify the denominator.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 9}$ as ${(x^{2} + 9)}^{\frac{1}{2}}$ .

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{1}{2}} {(x^{2} + 9)}^{\frac{1}{2}}}$

Step 5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{1}{2} + \frac{1}{2}}}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{1 + 1}{2}}}$

Step 5.4

Add $1$ and $1$ .

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{2}{2}}}$

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Cancel the common factor.

$\frac{f' (x)}{f (x)} = \frac{x}{{(x^{2} + 9)}^{\frac{2}{2}}}$

Step 5.5.2

Rewrite the expression.

$\frac{f' (x)}{f (x)} = \frac{x}{x^{2} + 9}$

Step 6

Evaluate the formula at $x = 4$ .

$\frac{4}{{({(4)}^{2} + 9)}^{1}}$

Step 7

Simplify the denominator.

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Step 7.1

Raise $4$ to the power of $2$ .

$\frac{4}{{(16 + 9)}^{1}}$

Step 7.2

Add $16$ and $9$ .

$\frac{4}{25^{1}}$

Step 7.3

Evaluate the exponent.

$\frac{4}{25}$

Step 8

Convert $\frac{4}{25}$ to a decimal.

$0.16$

Step 9

Convert $0.16$ to a percentage.

$16$ %