Calculus Examples

$2 \cot^{2} (π t + 2)$

Step 1

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

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

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 $g (t) = \cot (π t + 2)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cot (π t + 2)$ .

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

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 $n = 2$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\cot (π t + 2)$ .

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

Step 3

Multiply $2$ by $2$ .

$4 (\cot (π t + 2) \frac{d}{d t} [\cot (π t + 2)])$

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 $g (t) = π t + 2$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $π t + 2$ .

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

Step 4.2

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

$4 \cot (π t + 2) (- \csc^{2} (u_{2}) \frac{d}{d t} [π t + 2])$

Step 4.3

Replace all occurrences of $u_{2}$ with $π t + 2$ .

$4 \cot (π t + 2) (- \csc^{2} (π t + 2) \frac{d}{d t} [π t + 2])$

Step 5

Differentiate.

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Step 5.1

Multiply $- 1$ by $4$ .

$- 4 \cot (π t + 2) (\csc^{2} (π t + 2) \frac{d}{d t} [π t + 2])$

Step 5.2

By the Sum Rule, the derivative of $π t + 2$ with respect to $t$ is $\frac{d}{d t} [π t] + \frac{d}{d t} [2]$ .

$- 4 \cot (π t + 2) \csc^{2} (π t + 2) (\frac{d}{d t} [π t] + \frac{d}{d t} [2])$

Step 5.3

Since $π$ is constant with respect to $t$ , the derivative of $π t$ with respect to $t$ is $π \frac{d}{d t} [t]$ .

$- 4 \cot (π t + 2) \csc^{2} (π t + 2) (π \frac{d}{d t} [t] + \frac{d}{d t} [2])$

Step 5.4

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

$- 4 \cot (π t + 2) \csc^{2} (π t + 2) (π \cdot 1 + \frac{d}{d t} [2])$

Step 5.5

Multiply $π$ by $1$ .

$- 4 \cot (π t + 2) \csc^{2} (π t + 2) (π + \frac{d}{d t} [2])$

Step 5.6

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$- 4 \cot (π t + 2) \csc^{2} (π t + 2) (π + 0)$

Step 5.7

Simplify the expression.

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Step 5.7.1

Add $π$ and $0$ .

$- 4 \cot (π t + 2) \csc^{2} (π t + 2) π$

Step 5.7.2

Reorder the factors of $- 4 \cot (π t + 2) \csc^{2} (π t + 2) π$ .

$- 4 π \csc^{2} (π t + 2) \cot (π t + 2)$