Calculus Examples

x \ln (x)

$x \ln (x)$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x

$f (x) = x$ and

g (x) = \ln (x)

$g (x) = \ln (x)$ .

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

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

Step 2

The derivative of

\ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 3

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 = 1

$n = 1$ .

x \frac{1}{x} + \ln (x) \cdot 1

$x \frac{1}{x} + \ln (x) \cdot 1$

Step 4

Combine terms.

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Step 4.1

Combine

x

$x$ and

\frac{1}{x}

$\frac{1}{x}$ .

\frac{x}{x} + \ln (x) \cdot 1

$\frac{x}{x} + \ln (x) \cdot 1$

Step 4.2

Cancel the common factor of

x

$x$ .

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Step 4.2.1

Cancel the common factor.

\frac{x}{x} + \ln (x) \cdot 1

$\frac{x}{x} + \ln (x) \cdot 1$

Step 4.2.2

Rewrite the expression.

1 + \ln (x) \cdot 1

$1 + \ln (x) \cdot 1$

1 + \ln (x) \cdot 1

$1 + \ln (x) \cdot 1$

Step 4.3

Multiply

\ln (x)

$\ln (x)$ by

1

$1$ .

1 + \ln (x)

$1 + \ln (x)$

1 + \ln (x)

$1 + \ln (x)$

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