Calculus Examples

$\ln (| \cos (x) |)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = | \cos (x) |$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $| \cos (x) |$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [| \cos (x) |]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [| \cos (x) |]$

Step 1.3

Replace all occurrences of $u_{1}$ with $| \cos (x) |$ .

$\frac{1}{| \cos (x) |} \frac{d}{d x} [| \cos (x) |]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = \cos (x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$\frac{1}{| \cos (x) |} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [\cos (x)])$

Step 2.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{| \cos (x) |} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [\cos (x)])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$\frac{1}{| \cos (x) |} (\frac{\cos (x)}{| \cos (x) |} \frac{d}{d x} [\cos (x)])$

Step 3

Multiply $\frac{\cos (x)}{| \cos (x) |}$ by $\frac{1}{| \cos (x) |}$ .

$\frac{\cos (x)}{| \cos (x) | | \cos (x) |} \frac{d}{d x} [\cos (x)]$

Step 4

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{\cos (x)}{| \cos (x) \cos (x) |} \frac{d}{d x} [\cos (x)]$

Step 5

Raise $\cos (x)$ to the power of $1$ .

$\frac{\cos (x)}{| \cos^{1} (x) \cos (x) |} \frac{d}{d x} [\cos (x)]$

Step 6

Raise $\cos (x)$ to the power of $1$ .

$\frac{\cos (x)}{| \cos^{1} (x) \cos^{1} (x) |} \frac{d}{d x} [\cos (x)]$

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\cos (x)}{| {\cos (x)}^{1 + 1} |} \frac{d}{d x} [\cos (x)]$

Step 8

Add $1$ and $1$ .

$\frac{\cos (x)}{| \cos^{2} (x) |} \frac{d}{d x} [\cos (x)]$

Step 9

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{\cos (x)}{| \cos^{2} (x) |} (- \sin (x))$

Step 10

Combine $\frac{\cos (x)}{| \cos^{2} (x) |}$ and $\sin (x)$ .

$- \frac{\cos (x) \sin (x)}{| \cos^{2} (x) |}$

Step 11

Simplify.

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Step 11.1

Remove non-negative terms from the absolute value.

$- \frac{\cos (x) \sin (x)}{\cos^{2} (x)}$

Step 11.2

Cancel the common factor of $\cos (x)$ and $\cos^{2} (x)$ .

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Step 11.2.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

$- \frac{\cos (x) (\sin (x))}{\cos^{2} (x)}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$- \frac{\cos (x) (\sin (x))}{\cos (x) \cos (x)}$

Step 11.2.2.2

Cancel the common factor.

$- \frac{\cos (x) \sin (x)}{\cos (x) \cos (x)}$

Step 11.2.2.3

Rewrite the expression.

$- \frac{\sin (x)}{\cos (x)}$

Step 11.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$- \tan (x)$