Calculus Examples

$f (x) = 4 \ln (x) \cdot \tan (x)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (x) \tan (x)$ with respect to $x$ is $4 \frac{d}{d x} [\ln (x) \tan (x)]$ .

$4 \frac{d}{d x} [\ln (x) \tan (x)]$

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 $f (x) = \ln (x)$ and $g (x) = \tan (x)$ .

$4 (\ln (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\ln (x)])$

Step 3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$4 (\ln (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\ln (x)])$

Step 4

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

$4 (\ln (x) \sec^{2} (x) + \tan (x) \frac{1}{x})$

Step 5

Combine $\tan (x)$ and $\frac{1}{x}$ .

$4 (\ln (x) \sec^{2} (x) + \frac{\tan (x)}{x})$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$4 (\ln (x) \sec^{2} (x)) + 4 \frac{\tan (x)}{x}$

Step 6.2

Combine $4$ and $\frac{\tan (x)}{x}$ .

$4 \ln (x) \sec^{2} (x) + \frac{4 \tan (x)}{x}$

Step 6.3

Reorder terms.

$4 \sec^{2} (x) \ln (x) + \frac{4 \tan (x)}{x}$