Calculus Examples

$f (x) = 5 x \ln (x)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 x \ln (x)$ with respect to $x$ is $5 \frac{d}{d x} [x \ln (x)]$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

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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.

$5 \ln (x) + 5$