Calculus Examples

$\frac{π h^{2} (3 r - h)}{3}$

Step 1

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

$\frac{π}{3} \frac{d}{d h} [h^{2} (3 r - h)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d h} [f (h) g (h)]$ is $f (h) \frac{d}{d h} [g (h)] + g (h) \frac{d}{d h} [f (h)]$ where $f (h) = h^{2}$ and $g (h) = 3 r - h$ .

$\frac{π}{3} (h^{2} \frac{d}{d h} [3 r - h] + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 r - h$ with respect to $h$ is $\frac{d}{d h} [3 r] + \frac{d}{d h} [- h]$ .

$\frac{π}{3} (h^{2} (\frac{d}{d h} [3 r] + \frac{d}{d h} [- h]) + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.2

Since $3 r$ is constant with respect to $h$ , the derivative of $3 r$ with respect to $h$ is $0$ .

$\frac{π}{3} (h^{2} (0 + \frac{d}{d h} [- h]) + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.3

Add $0$ and $\frac{d}{d h} [- h]$ .

$\frac{π}{3} (h^{2} \frac{d}{d h} [- h] + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.4

Since $- 1$ is constant with respect to $h$ , the derivative of $- h$ with respect to $h$ is $- \frac{d}{d h} [h]$ .

$\frac{π}{3} (h^{2} (- \frac{d}{d h} [h]) + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.5

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

$\frac{π}{3} (h^{2} (- 1 \cdot 1) + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $- 1$ by $1$ .

$\frac{π}{3} (h^{2} \cdot - 1 + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.6.2

Move $- 1$ to the left of $h^{2}$ .

$\frac{π}{3} (- 1 \cdot h^{2} + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.6.3

Rewrite $- 1 h^{2}$ as $- h^{2}$ .

$\frac{π}{3} (- h^{2} + (3 r - h) \frac{d}{d h} [h^{2}])$

Step 3.7

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

$\frac{π}{3} (- h^{2} + (3 r - h) (2 h))$

Step 3.8

Move $2$ to the left of $3 r - h$ .

$\frac{π}{3} (- h^{2} + 2 \cdot (3 r - h) h)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{π}{3} (- h^{2} + (2 (3 r) + 2 (- h)) h)$

Step 4.2

Apply the distributive property.

$\frac{π}{3} (- h^{2} + 2 (3 r) h + 2 (- h) h)$

Step 4.3

Apply the distributive property.

$\frac{π}{3} (- h^{2}) + \frac{π}{3} (2 (3 r) h) + \frac{π}{3} (2 (- h) h)$

Step 4.4

Combine terms.

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Step 4.4.1

Combine $\frac{π}{3}$ and $h^{2}$ .

$- \frac{π h^{2}}{3} + \frac{π}{3} (2 (3 r) h) + \frac{π}{3} (2 (- h) h)$

Step 4.4.2

Multiply $3$ by $2$ .

$- \frac{π h^{2}}{3} + \frac{π}{3} (6 r h) + \frac{π}{3} (2 (- h) h)$

Step 4.4.3

Combine $6$ and $\frac{π}{3}$ .

$- \frac{π h^{2}}{3} + \frac{6 π}{3} (r h) + \frac{π}{3} (2 (- h) h)$

Step 4.4.4

Combine $r$ and $\frac{6 π}{3}$ .

$- \frac{π h^{2}}{3} + \frac{r (6 π)}{3} h + \frac{π}{3} (2 (- h) h)$

Step 4.4.5

Combine $\frac{r (6 π)}{3}$ and $h$ .

$- \frac{π h^{2}}{3} + \frac{r (6 π) h}{3} + \frac{π}{3} (2 (- h) h)$

Step 4.4.6

Move $6$ to the left of $r$ .

$- \frac{π h^{2}}{3} + \frac{6 \cdot r π h}{3} + \frac{π}{3} (2 (- h) h)$

Step 4.4.7

Cancel the common factor of $6$ and $3$ .

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Step 4.4.7.1

Factor $3$ out of $6 r π h$ .

$- \frac{π h^{2}}{3} + \frac{3 (2 r π h)}{3} + \frac{π}{3} (2 (- h) h)$

Step 4.4.7.2

Cancel the common factors.

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Step 4.4.7.2.1

Factor $3$ out of $3$ .

$- \frac{π h^{2}}{3} + \frac{3 (2 r π h)}{3 (1)} + \frac{π}{3} (2 (- h) h)$

Step 4.4.7.2.2

Cancel the common factor.

$- \frac{π h^{2}}{3} + \frac{3 (2 r π h)}{3 \cdot 1} + \frac{π}{3} (2 (- h) h)$

Step 4.4.7.2.3

Rewrite the expression.

$- \frac{π h^{2}}{3} + \frac{2 r π h}{1} + \frac{π}{3} (2 (- h) h)$

Step 4.4.7.2.4

Divide $2 r π h$ by $1$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{π}{3} (2 (- h) h)$

Step 4.4.8

Multiply $- 1$ by $2$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{π}{3} (- 2 h \cdot h)$

Step 4.4.9

Raise $h$ to the power of $1$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{π}{3} (- 2 (h^{1} h))$

Step 4.4.10

Raise $h$ to the power of $1$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{π}{3} (- 2 (h^{1} h^{1}))$

Step 4.4.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{π h^{2}}{3} + 2 r π h + \frac{π}{3} (- 2 h^{1 + 1})$

Step 4.4.12

Add $1$ and $1$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{π}{3} (- 2 h^{2})$

Step 4.4.13

Combine $- 2$ and $\frac{π}{3}$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{- 2 π}{3} h^{2}$

Step 4.4.14

Combine $\frac{- 2 π}{3}$ and $h^{2}$ .

$- \frac{π h^{2}}{3} + 2 r π h + \frac{- 2 π h^{2}}{3}$

Step 4.4.15

Move the negative in front of the fraction.

$- \frac{π h^{2}}{3} + 2 r π h - \frac{2 π h^{2}}{3}$

Step 4.4.16

Combine the numerators over the common denominator.

$2 r π h + \frac{- π h^{2} - 2 π h^{2}}{3}$

Step 4.4.17

Subtract $2 π h^{2}$ from $- π h^{2}$ .

$2 r π h + \frac{- 3 π h^{2}}{3}$

Step 4.4.18

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

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Step 4.4.18.1

Factor $3$ out of $- 3 π h^{2}$ .

$2 r π h + \frac{3 (- π h^{2})}{3}$

Step 4.4.18.2

Cancel the common factors.

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Step 4.4.18.2.1

Factor $3$ out of $3$ .

$2 r π h + \frac{3 (- π h^{2})}{3 (1)}$

Step 4.4.18.2.2

Cancel the common factor.

$2 r π h + \frac{3 (- π h^{2})}{3 \cdot 1}$

Step 4.4.18.2.3

Rewrite the expression.

$2 r π h + \frac{- π h^{2}}{1}$

Step 4.4.18.2.4

Divide $- π h^{2}$ by $1$ .

$2 r π h - π h^{2}$

Step 4.5

Reorder terms.

$2 π h r - π h^{2}$