Calculus Examples

$\ln (| \cos (\frac{π}{x}) |)$

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 $g (x) = | \cos (\frac{π}{x}) |$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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 $g (x) = \cos (\frac{π}{x})$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Raise $\cos (\frac{π}{x})$ to the power of $1$ .

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

Step 6

Raise $\cos (\frac{π}{x})$ to the power of $1$ .

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

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 $g (x) = \frac{π}{x}$ .

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Step 9.1

To apply the Chain Rule, set $u_{3}$ as $\frac{π}{x}$ .

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

Step 9.2

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

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

Step 9.3

Replace all occurrences of $u_{3}$ with $\frac{π}{x}$ .

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

Step 10

Differentiate.

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Step 10.1

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

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

Step 10.2

Since $π$ is constant with respect to $x$ , the derivative of $\frac{π}{x}$ with respect to $x$ is $π \frac{d}{d x} [\frac{1}{x}]$ .

$- \frac{\cos (\frac{π}{x}) \sin (\frac{π}{x})}{| \cos^{2} (\frac{π}{x}) |} (π \frac{d}{d x} [\frac{1}{x}])$

Step 10.3

Combine fractions.

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Step 10.3.1

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

$- \frac{π (\cos (\frac{π}{x}) \sin (\frac{π}{x}))}{| \cos^{2} (\frac{π}{x}) |} \frac{d}{d x} [\frac{1}{x}]$

Step 10.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$- \frac{π \cos (\frac{π}{x}) \sin (\frac{π}{x})}{| \cos^{2} (\frac{π}{x}) |} \frac{d}{d x} [x^{- 1}]$

Step 10.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 1$ .

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

Step 10.5

Combine fractions.

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Step 10.5.1

Multiply $- 1$ by $- 1$ .

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

Step 10.5.2

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

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

Step 10.5.3

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

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

Step 10.5.4

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11

Simplify.

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Step 11.1

Reorder terms.

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

Step 11.2

Remove non-negative terms from the absolute value.

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

Step 11.3

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

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Step 11.3.1

Factor $\cos (\frac{π}{x})$ out of $π \cos (\frac{π}{x}) \sin (\frac{π}{x})$ .

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

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

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

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

Step 11.3.2.2

Cancel the common factor.

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

Step 11.3.2.3

Rewrite the expression.

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

Step 11.4

Separate fractions.

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

Step 11.5

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

$\frac{π}{x^{2}} \tan (\frac{π}{x})$

Step 11.6

Combine $\frac{π}{x^{2}}$ and $\tan (\frac{π}{x})$ .

$\frac{π \tan (\frac{π}{x})}{x^{2}}$