Calculus Examples

$y = \cot^{2} (1 - \sin (x))$

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 $g (x) = \cot (1 - \sin (x))$ .

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Step 1.1

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

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

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 $n = 2$ .

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

Step 1.3

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

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

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 $g (x) = 1 - \sin (x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $1 - \sin (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $1 - \sin (x)$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

By the Sum Rule, the derivative of $1 - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \sin (x)]$ .

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

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 2 \cot (1 - \sin (x)) \csc^{2} (1 - \sin (x)) (0 + \frac{d}{d x} [- \sin (x)])$

Step 3.4

Add $0$ and $\frac{d}{d x} [- \sin (x)]$ .

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

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $- 2$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Reorder the factors of $2 \cot (1 - \sin (x)) \csc^{2} (1 - \sin (x)) \cos (x)$ .

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

Step 5.2

Rewrite $\csc (1 - \sin (x))$ in terms of sines and cosines.

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

Step 5.3

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

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

Step 5.4

One to any power is one.

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Rewrite $\cot (1 - \sin (x))$ in terms of sines and cosines.

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

Step 5.8

Combine.

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

Step 5.9

Multiply $\sin^{2} (1 - \sin (x))$ by $\sin (1 - \sin (x))$ by adding the exponents.

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Step 5.9.1

Multiply $\sin^{2} (1 - \sin (x))$ by $\sin (1 - \sin (x))$ .

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Step 5.9.1.1

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

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

Step 5.9.1.2

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

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

Step 5.9.2

Add $2$ and $1$ .

$\frac{2 \cos (x) \cos (1 - \sin (x))}{\sin^{3} (1 - \sin (x))}$

Step 5.10

Factor $\sin (1 - \sin (x))$ out of $\sin^{3} (1 - \sin (x))$ .

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

Step 5.11

Separate fractions.

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

Step 5.12

Rewrite $\frac{\cos (x)}{\sin (1 - \sin (x))}$ as a product.

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

Step 5.13

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

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

Step 5.14

Simplify.

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Step 5.14.1

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

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

Step 5.14.2

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

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

Step 5.15

Factor $\sin (1 - \sin (x))$ out of $\sin^{2} (1 - \sin (x))$ .

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

Step 5.16

Separate fractions.

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

Step 5.17

Convert from $\frac{\cos (1 - \sin (x))}{\sin (1 - \sin (x))}$ to $\cot (1 - \sin (x))$ .

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

Step 5.18

Separate fractions.

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

Step 5.19

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

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

Step 5.20

Divide $2$ by $1$ .

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

Step 5.21

Multiply $2 \csc (1 - \sin (x)) \cot (1 - \sin (x)) (\cos (x) \csc (1 - \sin (x)))$ .

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Step 5.21.1

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

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

Step 5.21.2

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

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

Step 5.21.3

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

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

Step 5.21.4

Add $1$ and $1$ .

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