Calculus Examples

5 x ln (x)

$5 x \ln (x)$

Step 1

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5 x ln (x)

$5 x \ln (x)$ with respect to

x

$x$ is

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

$5 \frac{d}{d x} [x \ln (x)]$ .

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

$5 \frac{d}{d x} [x \ln (x)]$

Step 2

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)

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

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

$5 (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])$

Step 3

The derivative of

ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

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

$5 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Combine

x

$x$ and

\frac{1}{x}

$\frac{1}{x}$ .

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

$5 (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 4.2

Cancel the common factor of

x

$x$ .

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Step 4.2.1

Cancel the common factor.

$5 (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 4.2.2

Rewrite the expression.

$5 (1 + \ln (x) \frac{d}{d x} [x])$

Step 4.3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$5 (1 + \ln (x) \cdot 1)$

Step 4.4

Multiply

$\ln (x)$ by

$1$ .

$5 (1 + \ln (x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$5 \cdot 1 + 5 \ln (x)$

Step 5.2

Multiply

$5$ by

$1$ .

$5 + 5 \ln (x)$

Step 5.3

Reorder terms.

$5 \ln (x) + 5$