Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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Step 9.1

To apply the Chain Rule, set $u_{3}$ as $\ln (x)$ .

$\frac{\cos (\ln (x))}{| \cos^{2} (\ln (x)) |} (\frac{d}{d u_{3}} [\cos (u_{3})] \frac{d}{d x} [\ln (x)])$

Step 9.2

The derivative of $\cos (u_{3})$ with respect to $u_{3}$ is $- \sin (u_{3})$ .

$\frac{\cos (\ln (x))}{| \cos^{2} (\ln (x)) |} (- \sin (u_{3}) \frac{d}{d x} [\ln (x)])$

Step 9.3

Replace all occurrences of $u_{3}$ with $\ln (x)$ .

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

Step 10

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

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

Step 11

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

$- \frac{\cos (\ln (x)) \sin (\ln (x))}{| \cos^{2} (\ln (x)) |} \cdot \frac{1}{x}$

Step 12

Multiply $\frac{1}{x}$ by $\frac{\cos (\ln (x)) \sin (\ln (x))}{| \cos^{2} (\ln (x)) |}$ .

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

Step 13

Simplify.

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Step 13.1

Remove non-negative terms from the absolute value.

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

Step 13.2

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

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Step 13.2.1

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

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

Step 13.2.2

Cancel the common factors.

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Step 13.2.2.1

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

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

Step 13.2.2.2

Cancel the common factor.

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

Step 13.2.2.3

Rewrite the expression.

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

Step 13.3

Separate fractions.

$- (\frac{1}{x} \cdot \frac{\sin (\ln (x))}{\cos (\ln (x))})$

Step 13.4

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

$- (\frac{1}{x} \tan (\ln (x)))$

Step 13.5

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

$- \frac{\tan (\ln (x))}{x}$