Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $r$ with respect to $M$ is $r'$ .

$r'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

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 $M$ is $\frac{d}{d M} [\frac{c}{2}] + \frac{d}{d M} [- \frac{M}{3}]$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

Multiply $- 1$ by $1$ .

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.3

Combine $- \frac{M^{2}}{3}$ and $2 (\frac{c}{2} - \frac{M}{3}) M$ using a common denominator.

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Step 3.3.1

Reorder $- \frac{M^{2}}{3}$ and $2 M (- \frac{M}{3} + \frac{c}{2})$ .

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

Step 3.3.2

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

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

Step 3.3.3

Combine $2 M (- \frac{M}{3} + \frac{c}{2})$ and $\frac{3}{3}$ .

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

Step 3.3.4

Combine the numerators over the common denominator.

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

Step 3.4

Multiply $3$ by $2$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Simplify the numerator.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Cancel the common factor of $3$ .

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Step 3.5.2.1.1.1

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

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

Step 3.5.2.1.1.2

Factor $3$ out of $6 M$ .

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

Step 3.5.2.1.1.3

Cancel the common factor.

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

Step 3.5.2.1.1.4

Rewrite the expression.

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

Step 3.5.2.1.2

Multiply $- 1$ by $2$ .

$\frac{- 2 M \cdot M + 6 M \frac{c}{2} - M^{2}}{3}$

Step 3.5.2.1.3

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

$\frac{- 2 (M^{1} M) + 6 M \frac{c}{2} - M^{2}}{3}$

Step 3.5.2.1.4

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

$\frac{- 2 (M^{1} M^{1}) + 6 M \frac{c}{2} - M^{2}}{3}$

Step 3.5.2.1.5

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

$\frac{- 2 M^{1 + 1} + 6 M \frac{c}{2} - M^{2}}{3}$

Step 3.5.2.1.6

Add $1$ and $1$ .

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

Step 3.5.2.1.7

Cancel the common factor of $2$ .

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Step 3.5.2.1.7.1

Factor $2$ out of $6 M$ .

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

Step 3.5.2.1.7.2

Cancel the common factor.

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

Step 3.5.2.1.7.3

Rewrite the expression.

$\frac{- 2 M^{2} + 3 M c - M^{2}}{3}$

Step 3.5.2.2

Subtract $M^{2}$ from $- 2 M^{2}$ .

$\frac{- 3 M^{2} + 3 M c}{3}$

Step 3.5.3

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

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Step 3.5.3.1

Factor $3 M$ out of $- 3 M^{2}$ .

$\frac{3 M (- M) + 3 M c}{3}$

Step 3.5.3.2

Factor $3 M$ out of $3 M c$ .

$\frac{3 M (- M) + 3 M (c)}{3}$

Step 3.5.3.3

Factor $3 M$ out of $3 M (- M) + 3 M (c)$ .

$\frac{3 M (- M + c)}{3}$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{3 M (- M + c)}{3}$

Step 3.5.4.2

Divide $M (- M + c)$ by $1$ .

$M (- M + c)$

Step 3.5.5

Apply the distributive property.

$M (- M) + M c$

Step 3.5.6

Rewrite using the commutative property of multiplication.

$- M \cdot M + M c$

Step 3.5.7

Multiply $M$ by $M$ by adding the exponents.

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Step 3.5.7.1

Move $M$ .

$- (M \cdot M) + M c$

Step 3.5.7.2

Multiply $M$ by $M$ .

$- M^{2} + M c$

Step 4

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

$r' = - M^{2} + M c$

Step 5

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

$\frac{d r}{d M} = - M^{2} + M c$