Calculus Examples

${(20 \cos (100 t))}^{2}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Factor $20$ out of $20 \cos (100 t)$ .

$\frac{d}{d t} [{(20 (\cos (100 t)))}^{2}]$

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $20 (\cos (100 t))$ .

$\frac{d}{d t} [20^{2} \cos^{2} (100 t)]$

Step 1.2.2

Raise $20$ to the power of $2$ .

$\frac{d}{d t} [400 \cos^{2} (100 t)]$

Step 1.3

Since $400$ is constant with respect to $t$ , the derivative of $400 \cos^{2} (100 t)$ with respect to $t$ is $400 \frac{d}{d t} [\cos^{2} (100 t)]$ .

$400 \frac{d}{d t} [\cos^{2} (100 t)]$

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 $g (t) = \cos (100 t)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (100 t)$ .

$400 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d t} [\cos (100 t)])$

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 $n = 2$ .

$400 (2 u_{1} \frac{d}{d t} [\cos (100 t)])$

Step 2.3

Replace all occurrences of $u_{1}$ with $\cos (100 t)$ .

$400 (2 \cos (100 t) \frac{d}{d t} [\cos (100 t)])$

Step 3

Multiply $2$ by $400$ .

$800 (\cos (100 t) \frac{d}{d t} [\cos (100 t)])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 100 t$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $100 t$ .

$800 \cos (100 t) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d t} [100 t])$

Step 4.2

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

$800 \cos (100 t) (- \sin (u_{2}) \frac{d}{d t} [100 t])$

Step 4.3

Replace all occurrences of $u_{2}$ with $100 t$ .

$800 \cos (100 t) (- \sin (100 t) \frac{d}{d t} [100 t])$

Step 5

Differentiate.

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Step 5.1

Multiply $- 1$ by $800$ .

$- 800 \cos (100 t) (\sin (100 t) \frac{d}{d t} [100 t])$

Step 5.2

Since $100$ is constant with respect to $t$ , the derivative of $100 t$ with respect to $t$ is $100 \frac{d}{d t} [t]$ .

$- 800 \cos (100 t) \sin (100 t) (100 \frac{d}{d t} [t])$

Step 5.3

Multiply $100$ by $- 800$ .

$- 80000 \cos (100 t) \sin (100 t) \frac{d}{d t} [t]$

Step 5.4

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

$- 80000 \cos (100 t) \sin (100 t) \cdot 1$

Step 5.5

Multiply $- 80000$ by $1$ .

$- 80000 \cos (100 t) \sin (100 t)$