Calculus Examples

$R = 5 \tan^{2} (3 π t)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (R) = \frac{d}{d t} (5 \tan^{2} (3 π t))$

Step 2

The derivative of $R$ with respect to $t$ is $R'$ .

$R'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $5$ is constant with respect to $t$ , the derivative of $5 \tan^{2} (3 π t)$ with respect to $t$ is $5 \frac{d}{d t} [\tan^{2} (3 π t)]$ .

$5 \frac{d}{d t} [\tan^{2} (3 π t)]$

Step 3.2

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 $g (t) = \tan (3 π t)$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $\tan (3 π t)$ .

$5 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d t} [\tan (3 π t)])$

Step 3.2.2

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

$5 (2 u_{1} \frac{d}{d t} [\tan (3 π t)])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $\tan (3 π t)$ .

$5 (2 \tan (3 π t) \frac{d}{d t} [\tan (3 π t)])$

Step 3.3

Multiply $2$ by $5$ .

$10 (\tan (3 π t) \frac{d}{d t} [\tan (3 π t)])$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \tan (t)$ and $g (t) = 3 π t$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $3 π t$ .

$10 \tan (3 π t) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d t} [3 π t])$

Step 3.4.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$10 \tan (3 π t) (\sec^{2} (u_{2}) \frac{d}{d t} [3 π t])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $3 π t$ .

$10 \tan (3 π t) (\sec^{2} (3 π t) \frac{d}{d t} [3 π t])$

Step 3.5

Differentiate.

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Step 3.5.1

Since $3 π$ is constant with respect to $t$ , the derivative of $3 π t$ with respect to $t$ is $3 π \frac{d}{d t} [t]$ .

$10 \tan (3 π t) \sec^{2} (3 π t) (3 π \frac{d}{d t} [t])$

Step 3.5.2

Multiply $3$ by $10$ .

$30 \tan (3 π t) \sec^{2} (3 π t) (π \frac{d}{d t} [t])$

Step 3.5.3

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

$30 \tan (3 π t) \sec^{2} (3 π t) (π \cdot 1)$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Multiply $π$ by $1$ .

$30 \tan (3 π t) \sec^{2} (3 π t) π$

Step 3.5.4.2

Reorder the factors of $30 \tan (3 π t) \sec^{2} (3 π t) π$ .

$30 π \sec^{2} (3 π t) \tan (3 π t)$

Step 4

Reform the equation by setting the left side equal to the right side.

$R' = 30 π \sec^{2} (3 π t) \tan (3 π t)$

Step 5

Replace $R'$ with $\frac{d R}{d t}$ .

$\frac{d R}{d t} = 30 π \sec^{2} (3 π t) \tan (3 π t)$