Calculus Examples

$f (x) = \cot^{2} (\sin (x))$

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 $g (x) = \cot (\sin (x))$ .

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Step 1.1

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

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

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 $n = 2$ .

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

Step 1.3

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

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

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 $g (x) = \sin (x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Multiply $- 1$ by $2$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

One to any power is one.

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

Step 5.5

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

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

Step 5.6

Move the negative in front of the fraction.

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

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

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Step 5.10.1

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

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

Step 5.10.2

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

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Step 5.10.2.1

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

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Step 5.10.2.1.1

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

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

Step 5.10.2.1.2

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

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

Step 5.10.2.2

Add $1$ and $2$ .

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

Step 5.11

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

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

Step 5.12

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

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

Step 5.13

Separate fractions.

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

Step 5.14

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

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

Step 5.15

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

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

Step 5.16

Separate fractions.

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

Step 5.17

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

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

Step 5.18

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

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

Step 5.19

Simplify.

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Step 5.19.1

Divide $\cos (x)$ by $1$ .

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

Step 5.19.2

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

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

Step 5.20

Separate fractions.

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

Step 5.21

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

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

Step 5.22

Divide $2$ by $1$ .

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

Step 5.23

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

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Step 5.23.1

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

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

Step 5.23.2

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

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

Step 5.23.3

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

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

Step 5.23.4

Add $1$ and $1$ .

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

Step 5.24

Multiply $2$ by $- 1$ .

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