Calculus Examples

$v = \frac{4}{3} \cdot (p r^{3})$

Step 1

Remove parentheses.

$v' = \frac{4}{3} \cdot (p r^{3})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d p} v' = \frac{d}{d p} (\frac{4}{3} \cdot (p r^{3}))$

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 $\frac{4}{3}$ .

$\frac{d}{d p} [\frac{p \cdot 4}{3} \cdot r^{3}]$

Step 4.1.2

Combine $\frac{p \cdot 4}{3}$ and $r^{3}$ .

$\frac{d}{d p} [\frac{p \cdot 4 r^{3}}{3}]$

Step 4.1.3

Move $4$ to the left of $p$ .

$\frac{d}{d p} [\frac{4 \cdot p r^{3}}{3}]$

Step 4.2

Since $\frac{4 r^{3}}{3}$ is constant with respect to $p$ , the derivative of $\frac{4 p r^{3}}{3}$ with respect to $p$ is $\frac{4 r^{3}}{3} \frac{d}{d p} [p]$ .

$\frac{4 r^{3}}{3} \frac{d}{d p} [p]$

Step 4.3

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

$\frac{4 r^{3}}{3} \cdot 1$

Step 4.4

Multiply $\frac{4 r^{3}}{3}$ by $1$ .

$\frac{4 r^{3}}{3}$

Step 5

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

$v'' = \frac{4 r^{3}}{3}$

Step 6

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

$\frac{d v}{d p} = \frac{4 r^{3}}{3}$