Calculus Examples

$v = π r^{2} h$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d h} v' = \frac{d}{d h} (π r^{2} h)$

Step 2

The derivative of $v'$ with respect to $h$ is $v''$ .

$v''$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$π r^{2} \frac{d}{d h} [h]$

Step 3.2

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

$π r^{2} \cdot 1$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $π$ by $1$ .

$r^{2} \cdot π$

Step 3.3.2

Reorder the factors of $r^{2} π$ .

$π r^{2}$

Step 4

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

$v'' = π r^{2}$

Step 5

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

$\frac{d v}{d h} = π r^{2}$