Algebra 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.

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d V} [V^{n}]$ is $n V^{n - 1}$ 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 $π$ and $\frac{4}{3}$ .

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

Step 4.1.2

Combine fractions.

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Step 4.1.2.1

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

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

Step 4.1.2.2

Move $4$ to the left of $π$ .

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

Step 4.1.3

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

$\frac{4 π}{3} \frac{d}{d V} [r^{3}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d V} [f (g (V))]$ is $f' (g (V)) g' (V)$ where $f (V) = V^{3}$ and $g (V) = r$ .

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Step 4.2.1

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

$\frac{4 π}{3} (\frac{d}{d u} [u^{3}] \frac{d}{d V} [r])$

Step 4.2.2

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

$\frac{4 π}{3} (3 u^{2} \frac{d}{d V} [r])$

Step 4.2.3

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

$\frac{4 π}{3} (3 r^{2} \frac{d}{d V} [r])$

Step 4.3

Simplify terms.

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Step 4.3.1

Combine $3$ and $\frac{4 π}{3}$ .

$\frac{3 (4 π)}{3} (r^{2} \frac{d}{d V} [r])$

Step 4.3.2

Multiply $4$ by $3$ .

$\frac{12 π}{3} (r^{2} \frac{d}{d V} [r])$

Step 4.3.3

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

$\frac{r^{2} (12 π)}{3} \frac{d}{d V} [r]$

Step 4.3.4

Move $12$ to the left of $r^{2}$ .

$\frac{12 \cdot r^{2} π}{3} \frac{d}{d V} [r]$

Step 4.3.5

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

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Step 4.3.5.1

Factor $3$ out of $12 r^{2} π$ .

$\frac{3 (4 r^{2} π)}{3} \frac{d}{d V} [r]$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $3$ out of $3$ .

$\frac{3 (4 r^{2} π)}{3 (1)} \frac{d}{d V} [r]$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{3 (4 r^{2} π)}{3 \cdot 1} \frac{d}{d V} [r]$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{4 r^{2} π}{1} \frac{d}{d V} [r]$

Step 4.3.5.2.4

Divide $4 r^{2} π$ by $1$ .

$4 r^{2} π \frac{d}{d V} [r]$

Step 4.4

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

$4 r^{2} π r'$

Step 4.5

Reorder the factors of $4 r^{2} π r'$ .

$4 π r^{2} r'$

Step 5

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

$1 = 4 π r^{2} r'$

Step 6

Solve for $r'$ .

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Step 6.1

Rewrite the equation as $4 π r^{2} r' = 1$ .

$4 π r^{2} r' = 1$

Step 6.2

Divide each term in $4 π r^{2} r' = 1$ by $4 π r^{2}$ and simplify.

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Step 6.2.1

Divide each term in $4 π r^{2} r' = 1$ by $4 π r^{2}$ .

$\frac{4 π r^{2} r'}{4 π r^{2}} = \frac{1}{4 π r^{2}}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{4 π r^{2} r'}{4 π r^{2}} = \frac{1}{4 π r^{2}}$

Step 6.2.2.1.2

Rewrite the expression.

$\frac{π r^{2} r'}{π r^{2}} = \frac{1}{4 π r^{2}}$

Step 6.2.2.2

Cancel the common factor of $π$ .

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Step 6.2.2.2.1

Cancel the common factor.

$\frac{π r^{2} r'}{π r^{2}} = \frac{1}{4 π r^{2}}$

Step 6.2.2.2.2

Rewrite the expression.

$\frac{r^{2} r'}{r^{2}} = \frac{1}{4 π r^{2}}$

Step 6.2.2.3

Cancel the common factor of $r^{2}$ .

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Step 6.2.2.3.1

Cancel the common factor.

$\frac{r^{2} r'}{r^{2}} = \frac{1}{4 π r^{2}}$

Step 6.2.2.3.2

Divide $r'$ by $1$ .

$r' = \frac{1}{4 π r^{2}}$

Step 7

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

$\frac{d r}{d V} = \frac{1}{4 π r^{2}}$