Calculus Examples

$f (x) = - 10 \ln (x) + 13 x^{2 - 11}$

Step 1

By the Sum Rule, the derivative of $- 10 \ln (x) + 13 x^{2 - 11}$ with respect to $x$ is $\frac{d}{d x} [- 10 \ln (x)] + \frac{d}{d x} [13 x^{2 - 11}]$ .

$\frac{d}{d x} [- 10 \ln (x)] + \frac{d}{d x} [13 x^{2 - 11}]$

Step 2

Evaluate $\frac{d}{d x} [- 10 \ln (x)]$ .

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Step 2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 \ln (x)$ with respect to $x$ is $- 10 \frac{d}{d x} [\ln (x)]$ .

$- 10 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [13 x^{2 - 11}]$

Step 2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$- 10 \frac{1}{x} + \frac{d}{d x} [13 x^{2 - 11}]$

Step 2.3

Combine $- 10$ and $\frac{1}{x}$ .

$\frac{- 10}{x} + \frac{d}{d x} [13 x^{2 - 11}]$

Step 2.4

Move the negative in front of the fraction.

$- \frac{10}{x} + \frac{d}{d x} [13 x^{2 - 11}]$

Step 3

Evaluate $\frac{d}{d x} [13 x^{2 - 11}]$ .

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Step 3.1

Subtract $11$ from $2$ .

$- \frac{10}{x} + \frac{d}{d x} [13 x^{- 9}]$

Step 3.2

Since $13$ is constant with respect to $x$ , the derivative of $13 x^{- 9}$ with respect to $x$ is $13 \frac{d}{d x} [x^{- 9}]$ .

$- \frac{10}{x} + 13 \frac{d}{d x} [x^{- 9}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 9$ .

$- \frac{10}{x} + 13 (- 9 x^{- 10})$

Step 3.4

Multiply $- 9$ by $13$ .

$- \frac{10}{x} - 117 x^{- 10}$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{10}{x} - 117 \frac{1}{x^{10}}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 117$ and $\frac{1}{x^{10}}$ .

$- \frac{10}{x} + \frac{- 117}{x^{10}}$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{10}{x} - \frac{117}{x^{10}}$