Calculus Examples

$y = 5 \cot^{2} (\sin (t))$

Step 1

Since $5$ is constant with respect to $t$ , the derivative of $5 \cot^{2} (\sin (t))$ with respect to $t$ is $5 \frac{d}{d t} [\cot^{2} (\sin (t))]$ .

$5 \frac{d}{d t} [\cot^{2} (\sin (t))]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = \cot (\sin (t))$ .

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Step 2.1

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

$5 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d t} [\cot (\sin (t))])$

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 $n = 2$ .

$5 (2 u_{1} \frac{d}{d t} [\cot (\sin (t))])$

Step 2.3

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

$5 (2 \cot (\sin (t)) \frac{d}{d t} [\cot (\sin (t))])$

Step 3

Multiply $2$ by $5$ .

$10 (\cot (\sin (t)) \frac{d}{d t} [\cot (\sin (t))])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cot (t)$ and $g (t) = \sin (t)$ .

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Step 4.1

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

$10 \cot (\sin (t)) (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d t} [\sin (t)])$

Step 4.2

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

$10 \cot (\sin (t)) (- \csc^{2} (u_{2}) \frac{d}{d t} [\sin (t)])$

Step 4.3

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

$10 \cot (\sin (t)) (- \csc^{2} (\sin (t)) \frac{d}{d t} [\sin (t)])$

Step 5

Multiply $- 1$ by $10$ .

$- 10 \cot (\sin (t)) (\csc^{2} (\sin (t)) \frac{d}{d t} [\sin (t)])$

Step 6

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

$- 10 \cot (\sin (t)) \csc^{2} (\sin (t)) \cos (t)$

Step 7

Simplify.

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Step 7.1

Reorder the factors of $- 10 \cot (\sin (t)) \csc^{2} (\sin (t)) \cos (t)$ .

$- 10 \csc^{2} (\sin (t)) \cos (t) \cot (\sin (t))$

Step 7.2

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

$- 10 {(\frac{1}{\sin (\sin (t))})}^{2} \cos (t) \cot (\sin (t))$

Step 7.3

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

$- 10 \frac{1^{2}}{\sin^{2} (\sin (t))} \cos (t) \cot (\sin (t))$

Step 7.4

One to any power is one.

$- 10 \frac{1}{\sin^{2} (\sin (t))} \cos (t) \cot (\sin (t))$

Step 7.5

Combine $- 10$ and $\frac{1}{\sin^{2} (\sin (t))}$ .

$\frac{- 10}{\sin^{2} (\sin (t))} \cos (t) \cot (\sin (t))$

Step 7.6

Move the negative in front of the fraction.

$- \frac{10}{\sin^{2} (\sin (t))} \cos (t) \cot (\sin (t))$

Step 7.7

Combine $\cos (t)$ and $\frac{10}{\sin^{2} (\sin (t))}$ .

$- \frac{\cos (t) \cdot 10}{\sin^{2} (\sin (t))} \cot (\sin (t))$

Step 7.8

Move $10$ to the left of $\cos (t)$ .

$- \frac{10 \cdot \cos (t)}{\sin^{2} (\sin (t))} \cot (\sin (t))$

Step 7.9

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

$- \frac{10 \cos (t)}{\sin^{2} (\sin (t))} \cdot \frac{\cos (\sin (t))}{\sin (\sin (t))}$

Step 7.10

Multiply $- \frac{10 \cos (t)}{\sin^{2} (\sin (t))} \cdot \frac{\cos (\sin (t))}{\sin (\sin (t))}$ .

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Step 7.10.1

Multiply $\frac{\cos (\sin (t))}{\sin (\sin (t))}$ by $\frac{10 \cos (t)}{\sin^{2} (\sin (t))}$ .

$- \frac{\cos (\sin (t)) (10 \cos (t))}{\sin (\sin (t)) \sin^{2} (\sin (t))}$

Step 7.10.2

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

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Step 7.10.2.1

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

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Step 7.10.2.1.1

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

$- \frac{\cos (\sin (t)) (10 \cos (t))}{\sin^{1} (\sin (t)) \sin^{2} (\sin (t))}$

Step 7.10.2.1.2

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

$- \frac{\cos (\sin (t)) (10 \cos (t))}{{\sin (\sin (t))}^{1 + 2}}$

Step 7.10.2.2

Add $1$ and $2$ .

$- \frac{\cos (\sin (t)) (10 \cos (t))}{\sin^{3} (\sin (t))}$

Step 7.11

Move $10$ to the left of $\cos (\sin (t))$ .

$- \frac{10 \cdot \cos (\sin (t)) \cos (t)}{\sin^{3} (\sin (t))}$

Step 7.12

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

$- \frac{10 \cos (\sin (t)) \cos (t)}{\sin (\sin (t)) \sin^{2} (\sin (t))}$

Step 7.13

Separate fractions.

$- (\frac{10 \cos (t)}{\sin^{2} (\sin (t))} \cdot \frac{\cos (\sin (t))}{\sin (\sin (t))})$

Step 7.14

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

$- (\frac{10 \cos (t)}{\sin^{2} (\sin (t))} \cot (\sin (t)))$

Step 7.15

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

$- (\frac{10 \cos (t)}{\sin (\sin (t)) \sin (\sin (t))} \cot (\sin (t)))$

Step 7.16

Separate fractions.

$- (\frac{10}{\sin (\sin (t))} \cdot \frac{\cos (t)}{\sin (\sin (t))} \cot (\sin (t)))$

Step 7.17

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

$- (\frac{10}{\sin (\sin (t))} (\cos (t) \frac{1}{\sin (\sin (t))}) \cot (\sin (t)))$

Step 7.18

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

$- (\frac{10}{\sin (\sin (t))} (\frac{\cos (t)}{1} \cdot \frac{1}{\sin (\sin (t))}) \cot (\sin (t)))$

Step 7.19

Simplify.

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Step 7.19.1

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

$- (\frac{10}{\sin (\sin (t))} (\cos (t) \frac{1}{\sin (\sin (t))}) \cot (\sin (t)))$

Step 7.19.2

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

$- (\frac{10}{\sin (\sin (t))} (\cos (t) \csc (\sin (t))) \cot (\sin (t)))$

Step 7.20

Separate fractions.

$- (\frac{10}{1} \cdot \frac{1}{\sin (\sin (t))} (\cos (t) \csc (\sin (t))) \cot (\sin (t)))$

Step 7.21

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

$- (\frac{10}{1} \csc (\sin (t)) (\cos (t) \csc (\sin (t))) \cot (\sin (t)))$

Step 7.22

Divide $10$ by $1$ .

$- (10 \csc (\sin (t)) (\cos (t) \csc (\sin (t))) \cot (\sin (t)))$

Step 7.23

Multiply $10 \csc (\sin (t)) (\cos (t) \csc (\sin (t)))$ .

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Step 7.23.1

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

$- (10 (\csc^{1} (\sin (t)) \csc (\sin (t))) \cos (t) \cot (\sin (t)))$

Step 7.23.2

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

$- (10 (\csc^{1} (\sin (t)) \csc^{1} (\sin (t))) \cos (t) \cot (\sin (t)))$

Step 7.23.3

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

$- (10 {\csc (\sin (t))}^{1 + 1} \cos (t) \cot (\sin (t)))$

Step 7.23.4

Add $1$ and $1$ .

$- (10 \csc^{2} (\sin (t)) \cos (t) \cot (\sin (t)))$

Step 7.24

Multiply $10$ by $- 1$ .

$- 10 \csc^{2} (\sin (t)) \cos (t) \cot (\sin (t))$