Calculus Examples

$y = \cot^{2} (\cos (θ))$

Step 1

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

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Step 1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d θ} [\cot (\cos (θ))]$

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 $n = 2$ .

$2 u_{1} \frac{d}{d θ} [\cot (\cos (θ))]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cot (\cos (θ))$ .

$2 \cot (\cos (θ)) \frac{d}{d θ} [\cot (\cos (θ))]$

Step 2

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

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Step 2.1

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

$2 \cot (\cos (θ)) (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d θ} [\cos (θ)])$

Step 2.2

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

$2 \cot (\cos (θ)) (- \csc^{2} (u_{2}) \frac{d}{d θ} [\cos (θ)])$

Step 2.3

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

$2 \cot (\cos (θ)) (- \csc^{2} (\cos (θ)) \frac{d}{d θ} [\cos (θ)])$

Step 3

Multiply $- 1$ by $2$ .

$- 2 \cot (\cos (θ)) (\csc^{2} (\cos (θ)) \frac{d}{d θ} [\cos (θ)])$

Step 4

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

$- 2 \cot (\cos (θ)) \csc^{2} (\cos (θ)) (- \sin (θ))$

Step 5

Multiply $- 1$ by $- 2$ .

$2 \cot (\cos (θ)) \csc^{2} (\cos (θ)) \sin (θ)$

Step 6

Simplify.

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Step 6.1

Reorder the factors of $2 \cot (\cos (θ)) \csc^{2} (\cos (θ)) \sin (θ)$ .

$2 \csc^{2} (\cos (θ)) \cot (\cos (θ)) \sin (θ)$

Step 6.2

Rewrite $\csc (\cos (θ))$ in terms of sines and cosines.

$2 {(\frac{1}{\sin (\cos (θ))})}^{2} \cot (\cos (θ)) \sin (θ)$

Step 6.3

Apply the product rule to $\frac{1}{\sin (\cos (θ))}$ .

$2 \frac{1^{2}}{\sin^{2} (\cos (θ))} \cot (\cos (θ)) \sin (θ)$

Step 6.4

One to any power is one.

$2 \frac{1}{\sin^{2} (\cos (θ))} \cot (\cos (θ)) \sin (θ)$

Step 6.5

Combine $2$ and $\frac{1}{\sin^{2} (\cos (θ))}$ .

$\frac{2}{\sin^{2} (\cos (θ))} \cot (\cos (θ)) \sin (θ)$

Step 6.6

Rewrite $\cot (\cos (θ))$ in terms of sines and cosines.

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

Step 6.7

Combine.

$\frac{2 \cos (\cos (θ))}{\sin^{2} (\cos (θ)) \sin (\cos (θ))} \sin (θ)$

Step 6.8

Multiply $\sin^{2} (\cos (θ))$ by $\sin (\cos (θ))$ by adding the exponents.

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Step 6.8.1

Multiply $\sin^{2} (\cos (θ))$ by $\sin (\cos (θ))$ .

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Step 6.8.1.1

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

$\frac{2 \cos (\cos (θ))}{\sin^{2} (\cos (θ)) \sin^{1} (\cos (θ))} \sin (θ)$

Step 6.8.1.2

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

$\frac{2 \cos (\cos (θ))}{{\sin (\cos (θ))}^{2 + 1}} \sin (θ)$

Step 6.8.2

Add $2$ and $1$ .

$\frac{2 \cos (\cos (θ))}{\sin^{3} (\cos (θ))} \sin (θ)$

Step 6.9

Combine $\frac{2 \cos (\cos (θ))}{\sin^{3} (\cos (θ))}$ and $\sin (θ)$ .

$\frac{2 \cos (\cos (θ)) \sin (θ)}{\sin^{3} (\cos (θ))}$

Step 6.10

Factor $\sin (\cos (θ))$ out of $\sin^{3} (\cos (θ))$ .

$\frac{2 \cos (\cos (θ)) \sin (θ)}{\sin (\cos (θ)) \sin^{2} (\cos (θ))}$

Step 6.11

Separate fractions.

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

Step 6.12

Convert from $\frac{\cos (\cos (θ))}{\sin (\cos (θ))}$ to $\cot (\cos (θ))$ .

$\frac{2 \sin (θ)}{\sin^{2} (\cos (θ))} \cot (\cos (θ))$

Step 6.13

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

$\frac{2 \sin (θ)}{\sin (\cos (θ)) \sin (\cos (θ))} \cot (\cos (θ))$

Step 6.14

Separate fractions.

$\frac{2}{\sin (\cos (θ))} \cdot \frac{\sin (θ)}{\sin (\cos (θ))} \cot (\cos (θ))$

Step 6.15

Rewrite $\frac{\sin (θ)}{\sin (\cos (θ))}$ as a product.

$\frac{2}{\sin (\cos (θ))} (\sin (θ) \frac{1}{\sin (\cos (θ))}) \cot (\cos (θ))$

Step 6.16

Write $\sin (θ)$ as a fraction with denominator $1$ .

$\frac{2}{\sin (\cos (θ))} (\frac{\sin (θ)}{1} \cdot \frac{1}{\sin (\cos (θ))}) \cot (\cos (θ))$

Step 6.17

Simplify.

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Step 6.17.1

Divide $\sin (θ)$ by $1$ .

$\frac{2}{\sin (\cos (θ))} (\sin (θ) \frac{1}{\sin (\cos (θ))}) \cot (\cos (θ))$

Step 6.17.2

Convert from $\frac{1}{\sin (\cos (θ))}$ to $\csc (\cos (θ))$ .

$\frac{2}{\sin (\cos (θ))} (\sin (θ) \csc (\cos (θ))) \cot (\cos (θ))$

Step 6.18

Separate fractions.

$\frac{2}{1} \cdot \frac{1}{\sin (\cos (θ))} (\sin (θ) \csc (\cos (θ))) \cot (\cos (θ))$

Step 6.19

Convert from $\frac{1}{\sin (\cos (θ))}$ to $\csc (\cos (θ))$ .

$\frac{2}{1} \csc (\cos (θ)) (\sin (θ) \csc (\cos (θ))) \cot (\cos (θ))$

Step 6.20

Divide $2$ by $1$ .

$2 \csc (\cos (θ)) (\sin (θ) \csc (\cos (θ))) \cot (\cos (θ))$

Step 6.21

Multiply $2 \csc (\cos (θ)) (\sin (θ) \csc (\cos (θ)))$ .

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Step 6.21.1

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

$2 (\csc^{1} (\cos (θ)) \csc (\cos (θ))) \sin (θ) \cot (\cos (θ))$

Step 6.21.2

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

$2 (\csc^{1} (\cos (θ)) \csc^{1} (\cos (θ))) \sin (θ) \cot (\cos (θ))$

Step 6.21.3

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

$2 {\csc (\cos (θ))}^{1 + 1} \sin (θ) \cot (\cos (θ))$

Step 6.21.4

Add $1$ and $1$ .

$2 \csc^{2} (\cos (θ)) \sin (θ) \cot (\cos (θ))$