Calculus Examples

$\frac{Gm M}{r^{2}}$

Step 1

Since $Gm M$ is constant with respect to $r$ , the derivative of $\frac{Gm M}{r^{2}}$ with respect to $r$ is $Gm M \frac{d}{d r} [\frac{1}{r^{2}}]$ .

$Gm M \frac{d}{d r} [\frac{1}{r^{2}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{r^{2}}$ as ${(r^{2})}^{- 1}$ .

$Gm M \frac{d}{d r} [{(r^{2})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(r^{2})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$Gm M \frac{d}{d r} [r^{2 \cdot - 1}]$

Step 2.2.2

Multiply $2$ by $- 1$ .

$Gm M \frac{d}{d r} [r^{- 2}]$

Step 3

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

$Gm M (- 2 r^{- 3})$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$Gm M (- 2 \frac{1}{r^{3}})$

Step 4.2

Combine terms.

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Step 4.2.1

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

$Gm M \frac{- 2}{r^{3}}$

Step 4.2.2

Move the negative in front of the fraction.

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

Step 4.2.3

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

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

Step 4.2.4

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

$- (\frac{M \cdot 2}{r^{3}} Gm)$

Step 4.2.5

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

$- \frac{2 M}{r^{3}} Gm$

Step 4.3

Reorder the factors of $- \frac{2 M}{r^{3}} Gm$ .

$- Gm \frac{2 M}{r^{3}}$

Step 4.4

Simplify.

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Step 4.4.1

Combine $- Gm$ and $\frac{2 M}{r^{3}}$ .

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

Step 4.4.2

Multiply $- 1$ by $2$ .

$\frac{- 2 M}{r^{3}} Gm$

Step 4.4.3

Move the negative in front of the fraction.

$- \frac{2 M}{r^{3}} Gm$