Calculus Examples

$v = \frac{1}{3} π r^{2} h$

Step 1

Combine fractions.

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Step 1.1

Combine $\frac{1}{3}$ and $π$ .

$\frac{d}{d r} [\frac{π}{3} r^{2} h]$

Step 1.2

Combine $\frac{π}{3}$ and $r^{2}$ .

$\frac{d}{d r} [\frac{π r^{2}}{3} h]$

Step 1.3

Combine $\frac{π r^{2}}{3}$ and $h$ .

$\frac{d}{d r} [\frac{π r^{2} h}{3}]$

Step 2

Since $\frac{π h}{3}$ is constant with respect to $r$ , the derivative of $\frac{π r^{2} h}{3}$ with respect to $r$ is $\frac{π h}{3} \frac{d}{d r} [r^{2}]$ .

$\frac{π h}{3} \frac{d}{d r} [r^{2}]$

Step 3

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

$\frac{π h}{3} (2 r)$

Step 4

Combine fractions.

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Step 4.1

Combine $2$ and $\frac{π h}{3}$ .

$\frac{2 (π h)}{3} r$

Step 4.2

Combine $\frac{2 (π h)}{3}$ and $r$ .

$\frac{2 π h r}{3}$