Calculus Examples

$y = \cot^{2} (\sin (θ))$

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 $g (θ) = \cot (\sin (θ))$ .

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Step 1.1

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

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

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 $n = 2$ .

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

Step 1.3

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

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

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 $g (θ) = \sin (θ)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Multiply $- 1$ by $2$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

One to any power is one.

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

Step 5.5

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

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

Step 5.6

Move the negative in front of the fraction.

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

Step 5.7

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

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

Step 5.8

Move $2$ to the left of $\cos (θ)$ .

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

Step 5.9

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

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

Step 5.10

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

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Step 5.10.1

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

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

Step 5.10.2

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

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Step 5.10.2.1

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

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Step 5.10.2.1.1

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

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

Step 5.10.2.1.2

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

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

Step 5.10.2.2

Add $1$ and $2$ .

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

Step 5.11

Move $2$ to the left of $\cos (\sin (θ))$ .

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

Step 5.12

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

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

Step 5.13

Separate fractions.

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

Step 5.14

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

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

Step 5.15

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

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

Step 5.16

Separate fractions.

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

Step 5.17

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

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

Step 5.18

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

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

Step 5.19

Simplify.

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Step 5.19.1

Divide $\cos (θ)$ by $1$ .

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

Step 5.19.2

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

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

Step 5.20

Separate fractions.

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

Step 5.21

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

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

Step 5.22

Divide $2$ by $1$ .

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

Step 5.23

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

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Step 5.23.1

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

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

Step 5.23.2

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

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

Step 5.23.3

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

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

Step 5.23.4

Add $1$ and $1$ .

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

Step 5.24

Multiply $2$ by $- 1$ .

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