Calculus Examples

$\frac{1}{3} \cdot (p r^{2} h)$

Step 1

Combine fractions.

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Step 1.1

Combine $p$ and $\frac{1}{3}$ .

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

Step 1.2

Combine $r^{2}$ and $\frac{p}{3}$ .

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

Step 1.3

Combine $\frac{r^{2} p}{3}$ and $h$ .

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

Step 2

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

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

Step 3

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

$\frac{p h}{3} (2 r)$

Step 4

Combine fractions.

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Step 4.1

Combine $2$ and $\frac{p h}{3}$ .

$\frac{2 (p h)}{3} r$

Step 4.2

Combine $\frac{2 (p h)}{3}$ and $r$ .

$\frac{2 (p h) r}{3}$

Step 4.3

Reorder terms.

$\frac{2 h p r}{3}$