Calculus Examples

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

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

Remove parentheses.

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

Step 1.2

Remove parentheses.

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

Step 2

Differentiate both sides of the equation.

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

Step 3

The derivative of $V$ with respect to $p$ 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 $p$ and $\frac{1}{3}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Multiply $\frac{r^{2} h}{3}$ by $1$ .

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

Step 5

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

$V' = \frac{r^{2} h}{3}$

Step 6

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

$\frac{d V}{d p} = \frac{r^{2} h}{3}$