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.

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

Step 1.2

Remove parentheses.

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Combine $p$ and $\frac{1}{3}$ .

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

Step 4.1.2

Combine fractions.

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.3

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

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

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (r'))]$ is $f' (g (r')) g' (r')$ where $f (r') = {r'}^{2}$ and $g (r') = r$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $r$ .

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

Step 4.2.2

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

$\frac{p h}{3} (2 u \frac{d}{d v} [r])$

Step 4.2.3

Replace all occurrences of $u$ with $r$ .

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

Step 4.3

Combine fractions.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move $2$ to the left of $r$ .

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

Step 4.4

Rewrite $\frac{d}{d v} [r]$ as $r'$ .

$\frac{2 r p h}{3} r'$

Step 4.5

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

$\frac{2 r p h r'}{3}$

Step 4.6

Reorder terms.

$\frac{2 h p r r'}{3}$

Step 5

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

$1 = \frac{2 h p r r'}{3}$

Step 6

Solve for $r'$ .

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Step 6.1

Rewrite the equation as $\frac{2 h p r r'}{3} = 1$ .

$\frac{2 h p r r'}{3} = 1$

Step 6.2

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 h p r r'}{3} = \frac{3}{2} \cdot 1$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{3}{2} \cdot \frac{2 h p r r'}{3}$ .

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Step 6.3.1.1.1

Cancel the common factor of $3$ .

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Step 6.3.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 h p r r'}{3} = \frac{3}{2} \cdot 1$

Step 6.3.1.1.1.2

Rewrite the expression.

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

Step 6.3.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.1.1.2.1

Factor $2$ out of $2 h p r r'$ .

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

Step 6.3.1.1.2.2

Cancel the common factor.

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

Step 6.3.1.1.2.3

Rewrite the expression.

$h p r r' = \frac{3}{2} \cdot 1$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply $\frac{3}{2}$ by $1$ .

$h p r r' = \frac{3}{2}$

Step 6.4

Divide each term in $h p r r' = \frac{3}{2}$ by $h p r$ and simplify.

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Step 6.4.1

Divide each term in $h p r r' = \frac{3}{2}$ by $h p r$ .

$\frac{h p r r'}{h p r} = \frac{\frac{3}{2}}{h p r}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $h$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{h p r r'}{h p r} = \frac{\frac{3}{2}}{h p r}$

Step 6.4.2.1.2

Rewrite the expression.

$\frac{p r r'}{p r} = \frac{\frac{3}{2}}{h p r}$

Step 6.4.2.2

Cancel the common factor of $p$ .

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Step 6.4.2.2.1

Cancel the common factor.

$\frac{p r r'}{p r} = \frac{\frac{3}{2}}{h p r}$

Step 6.4.2.2.2

Rewrite the expression.

$\frac{r r'}{r} = \frac{\frac{3}{2}}{h p r}$

Step 6.4.2.3

Cancel the common factor of $r$ .

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Step 6.4.2.3.1

Cancel the common factor.

$\frac{r r'}{r} = \frac{\frac{3}{2}}{h p r}$

Step 6.4.2.3.2

Divide $r'$ by $1$ .

$r' = \frac{\frac{3}{2}}{h p r}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$r' = \frac{3}{2} \cdot \frac{1}{h p r}$

Step 6.4.3.2

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

$r' = \frac{3}{2 h p r}$

Step 7

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

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