Calculus Examples

\frac{x^{3}}{3} - ln (x)

$\frac{x^{3}}{3} - \ln (x)$

Step 1

By the Sum Rule, the derivative of

\frac{x^{3}}{3} - ln (x)

$\frac{x^{3}}{3} - \ln (x)$ with respect to

x

$x$ is

\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [- ln (x)]

$\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [- \ln (x)]$ .

\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [- ln (x)]

$\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [- \ln (x)]$

Step 2

Evaluate

\frac{d}{d x} [\frac{x^{3}}{3}]

$\frac{d}{d x} [\frac{x^{3}}{3}]$ .

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Step 2.1

Since

\frac{1}{3}

$\frac{1}{3}$ is constant with respect to

x

$x$ , the derivative of

\frac{x^{3}}{3}

$\frac{x^{3}}{3}$ with respect to

x

$x$ is

\frac{1}{3} \frac{d}{d x} [x^{3}]

$\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- ln (x)]

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \ln (x)]$

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- ln (x)]

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- \ln (x)]$

Step 2.3

Combine

3

$3$ and

\frac{1}{3}

$\frac{1}{3}$ .

\frac{3}{3} x^{2} + \frac{d}{d x} [- ln (x)]

$\frac{3}{3} x^{2} + \frac{d}{d x} [- \ln (x)]$

Step 2.4

Combine

\frac{3}{3}

$\frac{3}{3}$ and

x^{2}

$x^{2}$ .

\frac{3 x^{2}}{3} + \frac{d}{d x} [- ln (x)]

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- \ln (x)]$

Step 2.5

Cancel the common factor of

3

$3$ .

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Step 2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- \ln (x)]$

Step 2.5.2

Divide

$x^{2}$ by

$1$ .

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

Step 3

Evaluate

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

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Step 3.1

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- \ln (x)$ with respect to

$x$ is

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

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

Step 3.2

The derivative of

$\ln (x)$ with respect to

$x$ is

$\frac{1}{x}$ .

$x^{2} - \frac{1}{x}$