Algebra Examples

$r = m^{2} (\frac{c}{2} - \frac{m}{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d r} (r) = \frac{d}{d r} (m^{2} (\frac{c}{2} - \frac{m}{3}))$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $\frac{c}{2} - \frac{m}{3}$ with respect to $r$ is $\frac{d}{d r} [\frac{c}{2}] + \frac{d}{d r} [- \frac{m}{3}]$ .

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

Step 3.2.2

Since $\frac{c}{2}$ is constant with respect to $r$ , the derivative of $\frac{c}{2}$ with respect to $r$ is $0$ .

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

Step 3.2.3

Add $0$ and $\frac{d}{d r} [- \frac{m}{3}]$ .

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

Step 3.2.4

Since $- \frac{1}{3}$ is constant with respect to $r$ , the derivative of $- \frac{m}{3}$ with respect to $r$ is $- \frac{1}{3} \frac{d}{d r} [m]$ .

$m^{2} (- \frac{1}{3} \frac{d}{d r} [m]) + (\frac{c}{2} - \frac{m}{3}) \frac{d}{d r} [m^{2}]$

Step 3.2.5

Combine $m^{2}$ and $\frac{1}{3}$ .

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

Step 3.3

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

$- \frac{m^{2}}{3} m' + (\frac{c}{2} - \frac{m}{3}) \frac{d}{d r} [m^{2}]$

Step 3.4

Combine $m'$ and $\frac{m^{2}}{3}$ .

$- \frac{m' m^{2}}{3} + (\frac{c}{2} - \frac{m}{3}) \frac{d}{d r} [m^{2}]$

Step 3.5

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

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Step 3.5.1

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

$- \frac{m' m^{2}}{3} + (\frac{c}{2} - \frac{m}{3}) (\frac{d}{d u} [u^{2}] \frac{d}{d r} [m])$

Step 3.5.2

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

$- \frac{m' m^{2}}{3} + (\frac{c}{2} - \frac{m}{3}) (2 u \frac{d}{d r} [m])$

Step 3.5.3

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

$- \frac{m' m^{2}}{3} + (\frac{c}{2} - \frac{m}{3}) (2 m \frac{d}{d r} [m])$

Step 3.6

Move $2$ to the left of $\frac{c}{2} - \frac{m}{3}$ .

$- \frac{m' m^{2}}{3} + 2 \cdot (\frac{c}{2} - \frac{m}{3}) m \frac{d}{d r} [m]$

Step 3.7

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

$- \frac{m' m^{2}}{3} + 2 (\frac{c}{2} - \frac{m}{3}) m m'$

Step 3.8

Reorder the factors of $2 (\frac{c}{2} - \frac{m}{3}) m m'$ .

$- \frac{m' m^{2}}{3} + 2 m m' (\frac{c}{2} - \frac{m}{3})$

Step 3.9

To write $2 m m' (\frac{c}{2} - \frac{m}{3})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{m' m^{2}}{3} + 2 m m' (\frac{c}{2} - \frac{m}{3}) \cdot \frac{3}{3}$

Step 3.10

Combine $2 m m' (\frac{c}{2} - \frac{m}{3})$ and $\frac{3}{3}$ .

$- \frac{m' m^{2}}{3} + \frac{2 m m' (\frac{c}{2} - \frac{m}{3}) \cdot 3}{3}$

Step 3.11

Combine the numerators over the common denominator.

$\frac{- m' m^{2} + 2 m m' (\frac{c}{2} - \frac{m}{3}) \cdot 3}{3}$

Step 3.12

Multiply $3$ by $2$ .

$\frac{- m' m^{2} + 6 m m' (\frac{c}{2} - \frac{m}{3})}{3}$

Step 3.13

Simplify.

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Step 3.13.1

Apply the distributive property.

$\frac{- m' m^{2} + 6 m m' \frac{c}{2} + 6 m m' (- \frac{m}{3})}{3}$

Step 3.13.2

Simplify the numerator.

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Step 3.13.2.1

Simplify each term.

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Step 3.13.2.1.1

Cancel the common factor of $2$ .

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Step 3.13.2.1.1.1

Factor $2$ out of $6 m m'$ .

$\frac{- m' m^{2} + 2 (3 m m') \frac{c}{2} + 6 m m' (- \frac{m}{3})}{3}$

Step 3.13.2.1.1.2

Cancel the common factor.

$\frac{- m' m^{2} + 2 (3 m m') \frac{c}{2} + 6 m m' (- \frac{m}{3})}{3}$

Step 3.13.2.1.1.3

Rewrite the expression.

$\frac{- m' m^{2} + 3 m m' c + 6 m m' (- \frac{m}{3})}{3}$

Step 3.13.2.1.2

Cancel the common factor of $3$ .

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Step 3.13.2.1.2.1

Move the leading negative in $- \frac{m}{3}$ into the numerator.

$\frac{- m' m^{2} + 3 m m' c + 6 m m' \frac{- m}{3}}{3}$

Step 3.13.2.1.2.2

Factor $3$ out of $6 m m'$ .

$\frac{- m' m^{2} + 3 m m' c + 3 (2 m m') \frac{- m}{3}}{3}$

Step 3.13.2.1.2.3

Cancel the common factor.

$\frac{- m' m^{2} + 3 m m' c + 3 (2 m m') \frac{- m}{3}}{3}$

Step 3.13.2.1.2.4

Rewrite the expression.

$\frac{- m' m^{2} + 3 m m' c + 2 m m' (- m)}{3}$

Step 3.13.2.1.3

Multiply $- 1$ by $2$ .

$\frac{- m' m^{2} + 3 m m' c - 2 m m' m}{3}$

Step 3.13.2.1.4

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

$\frac{- m' m^{2} + 3 m m' c - 2 (m^{1} m) m'}{3}$

Step 3.13.2.1.5

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

$\frac{- m' m^{2} + 3 m m' c - 2 (m^{1} m^{1}) m'}{3}$

Step 3.13.2.1.6

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

$\frac{- m' m^{2} + 3 m m' c - 2 m^{1 + 1} m'}{3}$

Step 3.13.2.1.7

Add $1$ and $1$ .

$\frac{- m' m^{2} + 3 m m' c - 2 m^{2} m'}{3}$

Step 3.13.2.2

Subtract $2 m^{2} m'$ from $- m' m^{2}$ .

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Step 3.13.2.2.1

Move $m'$ .

$\frac{- m^{2} m' - 2 m^{2} m' + 3 m m' c}{3}$

Step 3.13.2.2.2

Subtract $2 m^{2} m'$ from $- m^{2} m'$ .

$\frac{- 3 m^{2} m' + 3 m m' c}{3}$

Step 3.13.2.3

Reorder factors in $- 3 m^{2} m' + 3 m m' c$ .

$\frac{- 3 m^{2} m' + 3 m c m'}{3}$

Step 3.13.3

Reorder terms.

$\frac{- 3 m^{2} m' + 3 c m m'}{3}$

Step 3.13.4

Factor $3 m m'$ out of $- 3 m^{2} m' + 3 c m m'$ .

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Step 3.13.4.1

Factor $3 m m'$ out of $- 3 m^{2} m'$ .

$\frac{3 m m' (- m) + 3 c m m'}{3}$

Step 3.13.4.2

Factor $3 m m'$ out of $3 c m m'$ .

$\frac{3 m m' (- m) + 3 m m' (c)}{3}$

Step 3.13.4.3

Factor $3 m m'$ out of $3 m m' (- m) + 3 m m' (c)$ .

$\frac{3 m m' (- m + c)}{3}$

Step 3.13.5

Cancel the common factor of $3$ .

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Step 3.13.5.1

Cancel the common factor.

$\frac{3 m m' (- m + c)}{3}$

Step 3.13.5.2

Divide $m m' (- m + c)$ by $1$ .

$m m' (- m + c)$

Step 3.13.6

Apply the distributive property.

$m m' (- m) + m m' c$

Step 3.13.7

Rewrite using the commutative property of multiplication.

$- m m' m + m m' c$

Step 3.13.8

Multiply $m$ by $m$ by adding the exponents.

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Step 3.13.8.1

Move $m$ .

$- (m \cdot m) m' + m m' c$

Step 3.13.8.2

Multiply $m$ by $m$ .

$- m^{2} m' + m m' c$

Step 3.13.9

Reorder factors in $- m^{2} m' + m m' c$ .

$- m^{2} m' + m c m'$

Step 4

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

$1 = - m^{2} m' + m c m'$

Step 5

Solve for $m'$ .

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Step 5.1

Rewrite the equation as $- m^{2} m' + m c m' = 1$ .

$- m^{2} m' + m c m' = 1$

Step 5.2

Factor $m m'$ out of $- m^{2} m' + m c m'$ .

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Step 5.2.1

Factor $m m'$ out of $- m^{2} m'$ .

$m m' (- m) + m c m' = 1$

Step 5.2.2

Factor $m m'$ out of $m c m'$ .

$m m' (- m) + m m' c = 1$

Step 5.2.3

Factor $m m'$ out of $m m' (- m) + m m' c$ .

$m m' (- m + c) = 1$

Step 5.3

Divide each term in $m m' (- m + c) = 1$ by $m (- m + c)$ and simplify.

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Step 5.3.1

Divide each term in $m m' (- m + c) = 1$ by $m (- m + c)$ .

$\frac{m m' (- m + c)}{m (- m + c)} = \frac{1}{m (- m + c)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $m$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{m m' (- m + c)}{m (- m + c)} = \frac{1}{m (- m + c)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{m' (- m + c)}{- m + c} = \frac{1}{m (- m + c)}$

Step 5.3.2.2

Cancel the common factor of $- m + c$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{m' (- m + c)}{- m + c} = \frac{1}{m (- m + c)}$

Step 5.3.2.2.2

Divide $m'$ by $1$ .

$m' = \frac{1}{m (- m + c)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Factor $- 1$ out of $- m$ .

$m' = \frac{1}{m (- (m) + c)}$

Step 5.3.3.2

Factor $- 1$ out of $c$ .

$m' = \frac{1}{m (- (m) - 1 (- c))}$

Step 5.3.3.3

Factor $- 1$ out of $- (m) - 1 (- c)$ .

$m' = \frac{1}{m (- (m - c))}$

Step 5.3.3.4

Rewrite negatives.

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Step 5.3.3.4.1

Rewrite $- (m - c)$ as $- 1 (m - c)$ .

$m' = \frac{1}{m (- 1 (m - c))}$

Step 5.3.3.4.2

Move the negative in front of the fraction.

$m' = - \frac{1}{m (m - c)}$

Step 6

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

$\frac{d m}{d r} = - \frac{1}{m (m - c)}$