Calculus Examples

$V = \frac{1}{3} π r^{2} h$

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

Remove parentheses.

$V = \frac{1}{3} \cdot ((π r^{2}) h)$

Step 1.2

Remove parentheses.

$V = \frac{1}{3} \cdot (π r^{2} h)$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d r} (V) = \frac{d}{d r} (\frac{1}{3} \cdot (π r^{2} h))$

Step 3

The derivative of $V$ with respect to $r$ is $V'$ .

$V'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Combine fractions.

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Step 4.1.1

Combine $π$ and $\frac{1}{3}$ .

$\frac{d}{d r} [\frac{π}{3} \cdot (r^{2} h)]$

Step 4.1.2

Combine $r^{2}$ and $\frac{π}{3}$ .

$\frac{d}{d r} [\frac{r^{2} π}{3} \cdot h]$

Step 4.1.3

Combine $\frac{r^{2} π}{3}$ and $h$ .

$\frac{d}{d r} [\frac{r^{2} π h}{3}]$

Step 4.2

Since $\frac{π h}{3}$ is constant with respect to $r$ , the derivative of $\frac{r^{2} π h}{3}$ 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 4.3

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 4.4

Combine fractions.

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Step 4.4.1

Combine $2$ and $\frac{π h}{3}$ .

$\frac{2 (π h)}{3} r$

Step 4.4.2

Combine $\frac{2 (π h)}{3}$ and $r$ .

$\frac{2 π h r}{3}$

Step 5

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

$V' = \frac{2 π h r}{3}$

Step 6

Replace $V'$ with $\frac{d V}{d r}$ .

$\frac{d V}{d r} = \frac{2 π h r}{3}$