Calculus Examples

$y = 7 \cot^{4} (\frac{1}{7} x)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $\frac{1}{7}$ and $x$ .

$\frac{d}{d x} [7 \cot^{4} (\frac{x}{7})]$

Step 1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 \cot^{4} (\frac{x}{7})$ with respect to $x$ is $7 \frac{d}{d x} [\cot^{4} (\frac{x}{7})]$ .

$7 \frac{d}{d x} [\cot^{4} (\frac{x}{7})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \cot (\frac{x}{7})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cot (\frac{x}{7})$ .

$7 (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\cot (\frac{x}{7})])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$7 (4 {u_{1}}^{3} \frac{d}{d x} [\cot (\frac{x}{7})])$

Step 2.3

Replace all occurrences of $u_{1}$ with $\cot (\frac{x}{7})$ .

$7 (4 \cot^{3} (\frac{x}{7}) \frac{d}{d x} [\cot (\frac{x}{7})])$

Step 3

Multiply $4$ by $7$ .

$28 (\cot^{3} (\frac{x}{7}) \frac{d}{d x} [\cot (\frac{x}{7})])$

Step 4

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 $g (x) = \frac{x}{7}$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{7}$ .

$28 \cot^{3} (\frac{x}{7}) (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d x} [\frac{x}{7}])$

Step 4.2

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

$28 \cot^{3} (\frac{x}{7}) (- \csc^{2} (u_{2}) \frac{d}{d x} [\frac{x}{7}])$

Step 4.3

Replace all occurrences of $u_{2}$ with $\frac{x}{7}$ .

$28 \cot^{3} (\frac{x}{7}) (- \csc^{2} (\frac{x}{7}) \frac{d}{d x} [\frac{x}{7}])$

Step 5

Differentiate.

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Step 5.1

Multiply $- 1$ by $28$ .

$- 28 \cot^{3} (\frac{x}{7}) (\csc^{2} (\frac{x}{7}) \frac{d}{d x} [\frac{x}{7}])$

Step 5.2

Since $\frac{1}{7}$ is constant with respect to $x$ , the derivative of $\frac{x}{7}$ with respect to $x$ is $\frac{1}{7} \frac{d}{d x} [x]$ .

$- 28 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}) (\frac{1}{7} \frac{d}{d x} [x])$

Step 5.3

Simplify terms.

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Step 5.3.1

Combine $\frac{1}{7}$ and $- 28$ .

$\frac{- 28}{7} \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}) \frac{d}{d x} [x]$

Step 5.3.2

Combine $\frac{- 28}{7}$ and $\cot^{3} (\frac{x}{7})$ .

$\frac{- 28 \cot^{3} (\frac{x}{7})}{7} \csc^{2} (\frac{x}{7}) \frac{d}{d x} [x]$

Step 5.3.3

Combine $\frac{- 28 \cot^{3} (\frac{x}{7})}{7}$ and $\csc^{2} (\frac{x}{7})$ .

$\frac{- 28 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7})}{7} \frac{d}{d x} [x]$

Step 5.3.4

Cancel the common factor of $- 28$ and $7$ .

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Step 5.3.4.1

Factor $7$ out of $- 28 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7})$ .

$\frac{7 (- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}))}{7} \frac{d}{d x} [x]$

Step 5.3.4.2

Cancel the common factors.

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Step 5.3.4.2.1

Factor $7$ out of $7$ .

$\frac{7 (- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}))}{7 (1)} \frac{d}{d x} [x]$

Step 5.3.4.2.2

Cancel the common factor.

$\frac{7 (- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}))}{7 \cdot 1} \frac{d}{d x} [x]$

Step 5.3.4.2.3

Rewrite the expression.

$\frac{- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7})}{1} \frac{d}{d x} [x]$

Step 5.3.4.2.4

Divide $- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7})$ by $1$ .

$- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}) \frac{d}{d x} [x]$

Step 5.4

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

$- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7}) \cdot 1$

Step 5.5

Multiply $- 4$ by $1$ .

$- 4 \cot^{3} (\frac{x}{7}) \csc^{2} (\frac{x}{7})$