Calculus Examples

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

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

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

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 $g (t) = \cos (\cot (t))$ .

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

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

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

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

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

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

Multiply $2$ by $2$ .

$4 (\cos (\cot (t)) \frac{d}{d t} [\cos (\cot (t))])$

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

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

$4 \cos (\cot (t)) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d t} [\cot (t)])$

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$4 \cos (\cot (t)) (- \sin (u_{2}) \frac{d}{d t} [\cot (t)])$

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

$4 \cos (\cot (t)) (- \sin (\cot (t)) \frac{d}{d t} [\cot (t)])$

Multiply $- 1$ by $4$ .

$- 4 \cos (\cot (t)) (\sin (\cot (t)) \frac{d}{d t} [\cot (t)])$

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

$- 4 \cos (\cot (t)) \sin (\cot (t)) (- \csc^{2} (t))$

Multiply $- 1$ by $- 4$ .

$4 \cos (\cot (t)) \sin (\cot (t)) \csc^{2} (t)$

Simplify.

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Reorder the factors of $4 \cos (\cot (t)) \sin (\cot (t)) \csc^{2} (t)$ .

$4 \csc^{2} (t) \cos (\cot (t)) \sin (\cot (t))$

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

$4 {(\frac{1}{\sin (t)})}^{2} \cos (\cot (t)) \sin (\cot (t))$

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

$4 \frac{1^{2}}{\sin^{2} (t)} \cos (\cot (t)) \sin (\cot (t))$

One to any power is one.

$4 \frac{1}{\sin^{2} (t)} \cos (\cot (t)) \sin (\cot (t))$

Combine $4$ and $\frac{1}{\sin^{2} (t)}$ .

$\frac{4}{\sin^{2} (t)} \cos (\cot (t)) \sin (\cot (t))$

Combine $\frac{4}{\sin^{2} (t)}$ and $\cos (\cot (t))$ .

$\frac{4 \cos (\cot (t))}{\sin^{2} (t)} \sin (\cot (t))$

Combine $\frac{4 \cos (\cot (t))}{\sin^{2} (t)}$ and $\sin (\cot (t))$ .

$\frac{4 \cos (\cot (t)) \sin (\cot (t))}{\sin^{2} (t)}$

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

$\frac{4 \cos (\cot (t)) \sin (\cot (t))}{\sin (t) \sin (t)}$

Separate fractions.

$\frac{4 \sin (\cot (t))}{\sin (t)} \cdot \frac{\cos (\cot (t))}{\sin (t)}$

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

$\frac{4 \sin (\cot (t))}{\sin (t)} (\cos (\cot (t)) \frac{1}{\sin (t)})$

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

$\frac{4 \sin (\cot (t))}{\sin (t)} (\frac{\cos (\cot (t))}{1} \cdot \frac{1}{\sin (t)})$

Simplify.

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

$\frac{4 \sin (\cot (t))}{\sin (t)} (\cos (\cot (t)) \frac{1}{\sin (t)})$

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

$\frac{4 \sin (\cot (t))}{\sin (t)} (\cos (\cot (t)) \csc (t))$

Separate fractions.

$\frac{4}{1} \cdot \frac{\sin (\cot (t))}{\sin (t)} (\cos (\cot (t)) \csc (t))$

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

$\frac{4}{1} (\sin (\cot (t)) \frac{1}{\sin (t)}) (\cos (\cot (t)) \csc (t))$

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

$\frac{4}{1} (\frac{\sin (\cot (t))}{1} \cdot \frac{1}{\sin (t)}) (\cos (\cot (t)) \csc (t))$

Simplify.

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Divide $\sin (\cot (t))$ by $1$ .

$\frac{4}{1} (\sin (\cot (t)) \frac{1}{\sin (t)}) (\cos (\cot (t)) \csc (t))$

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

$\frac{4}{1} (\sin (\cot (t)) \csc (t)) (\cos (\cot (t)) \csc (t))$

Divide $4$ by $1$ .

$4 (\sin (\cot (t)) \csc (t)) (\cos (\cot (t)) \csc (t))$

Multiply $4 \sin (\cot (t)) \csc (t) (\cos (\cot (t)) \csc (t))$ .

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

$4 \sin (\cot (t)) (\csc^{1} (t) \csc (t)) \cos (\cot (t))$

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

$4 \sin (\cot (t)) (\csc^{1} (t) \csc^{1} (t)) \cos (\cot (t))$

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

$4 \sin (\cot (t)) {\csc (t)}^{1 + 1} \cos (\cot (t))$

Add $1$ and $1$ .

$4 \sin (\cot (t)) \csc^{2} (t) \cos (\cot (t))$