Calculus Examples

$v = \frac{4}{3} π r^{3}$

Step 1

Remove parentheses.

$v' = \frac{4}{3} \cdot (π r^{3})$

Step 2

Differentiate both sides of the equation.

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

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 $\frac{4}{3}$ .

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

Step 4.1.2

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

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

Step 4.1.3

Move $4$ to the left of $π$ .

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

Step 4.2

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

$\frac{4 π}{3} \frac{d}{d r} [r^{3}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 3$ .

$\frac{4 π}{3} (3 r^{2})$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $3$ and $\frac{4 π}{3}$ .

$\frac{3 (4 π)}{3} r^{2}$

Step 4.4.2

Multiply $4$ by $3$ .

$\frac{12 π}{3} r^{2}$

Step 4.4.3

Combine $\frac{12 π}{3}$ and $r^{2}$ .

$\frac{12 π r^{2}}{3}$

Step 4.4.4

Cancel the common factor of $12$ and $3$ .

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Step 4.4.4.1

Factor $3$ out of $12 π r^{2}$ .

$\frac{3 (4 π r^{2})}{3}$

Step 4.4.4.2

Cancel the common factors.

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Step 4.4.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (4 π r^{2})}{3 (1)}$

Step 4.4.4.2.2

Cancel the common factor.

$\frac{3 (4 π r^{2})}{3 \cdot 1}$

Step 4.4.4.2.3

Rewrite the expression.

$\frac{4 π r^{2}}{1}$

Step 4.4.4.2.4

Divide $4 π r^{2}$ by $1$ .

$4 π r^{2}$

Step 5

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

$v'' = 4 π r^{2}$

Step 6

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

$\frac{d v}{d r} = 4 π r^{2}$