Calculus Examples

$y = \cot^{2} (\frac{1}{3 x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cot (\frac{1}{3 x})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cot (\frac{1}{3 x})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cot (\frac{1}{3 x})]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d x} [\cot (\frac{1}{3 x})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cot (\frac{1}{3 x})$ .

$2 \cot (\frac{1}{3 x}) \frac{d}{d x} [\cot (\frac{1}{3 x})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cot (x)$ and $g (x) = \frac{1}{3 x}$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{1}{3 x}$ .

$2 \cot (\frac{1}{3 x}) (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d x} [\frac{1}{3 x}])$

Step 2.2

The derivative of $\cot (u_{2})$ with respect to $u_{2}$ is $- \csc^{2} (u_{2})$ .

$2 \cot (\frac{1}{3 x}) (- \csc^{2} (u_{2}) \frac{d}{d x} [\frac{1}{3 x}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\frac{1}{3 x}$ .

$2 \cot (\frac{1}{3 x}) (- \csc^{2} (\frac{1}{3 x}) \frac{d}{d x} [\frac{1}{3 x}])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

$- 2 \cot (\frac{1}{3 x}) (\csc^{2} (\frac{1}{3 x}) \frac{d}{d x} [\frac{1}{3 x}])$

Step 3.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3 x}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{1}{x}]$ .

$- 2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x}) (\frac{1}{3} \frac{d}{d x} [\frac{1}{x}])$

Step 3.3

Combine fractions.

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Step 3.3.1

Combine $\frac{1}{3}$ and $- 2$ .

$\frac{- 2}{3} \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x}) \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.2

Combine $\frac{- 2}{3}$ and $\cot (\frac{1}{3 x})$ .

$\frac{- 2 \cot (\frac{1}{3 x})}{3} \csc^{2} (\frac{1}{3 x}) \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.3

Combine $\frac{- 2 \cot (\frac{1}{3 x})}{3}$ and $\csc^{2} (\frac{1}{3 x})$ .

$\frac{- 2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3} \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Move the negative in front of the fraction.

$- \frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3} \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.4.2

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

$- \frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3} \frac{d}{d x} [x^{- 1}]$

Step 3.4

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

$- \frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3} (- x^{- 2})$

Step 3.5

Combine fractions.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

$1 \frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3} x^{- 2}$

Step 3.5.2

Multiply $\frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3}$ by $1$ .

$\frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3} x^{- 2}$

Step 3.5.3

Combine $\frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3}$ and $x^{- 2}$ .

$\frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x}) x^{- 2}}{3}$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

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

$\frac{2 \cot (\frac{1}{3 x}) \csc^{2} (\frac{1}{3 x})}{3 x^{2}}$

Step 3.5.4.2

Reorder terms.

$\frac{2 \csc^{2} (\frac{1}{3 x}) \cot (\frac{1}{3 x})}{3 x^{2}}$