Calculus Examples

$- \frac{A}{r^{2}} + \frac{B}{r^{3}}$

Step 1

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

$\frac{d}{d r} [- \frac{A}{r^{2}}] + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2

Evaluate $\frac{d}{d r} [- \frac{A}{r^{2}}]$ .

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Step 2.1

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

$- A \frac{d}{d r} [\frac{1}{r^{2}}] + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.2

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

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = r^{- 1}$ and $g (r) = r^{2}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $r^{2}$ .

$- A (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d r} [r^{2}]) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.3.2

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

$- A (- {u_{1}}^{- 2} \frac{d}{d r} [r^{2}]) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $r^{2}$ .

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

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

$- A (- r^{2 \cdot - 2} (2 r)) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.5.2

Multiply $2$ by $- 2$ .

$- A (- r^{- 4} (2 r)) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.6

Multiply $2$ by $- 1$ .

$- A (- 2 r^{- 4} r) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.7

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

$- A (- 2 (r^{1} r^{- 4})) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.8

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

$- A (- 2 r^{1 - 4}) + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 2.9

Subtract $4$ from $1$ .

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

Step 2.10

Multiply $- 2$ by $- 1$ .

$2 A r^{- 3} + \frac{d}{d r} [\frac{B}{r^{3}}]$

Step 3

Evaluate $\frac{d}{d r} [\frac{B}{r^{3}}]$ .

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Step 3.1

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

$2 A r^{- 3} + B \frac{d}{d r} [\frac{1}{r^{3}}]$

Step 3.2

Rewrite $\frac{1}{r^{3}}$ as ${(r^{3})}^{- 1}$ .

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = r^{- 1}$ and $g (r) = r^{3}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $r^{3}$ .

$2 A r^{- 3} + B (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d r} [r^{3}])$

Step 3.3.2

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

$2 A r^{- 3} + B (- {u_{2}}^{- 2} \frac{d}{d r} [r^{3}])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $r^{3}$ .

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

Step 3.4

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

$2 A r^{- 3} + B (- {(r^{3})}^{- 2} (3 r^{2}))$

Step 3.5

Multiply the exponents in ${(r^{3})}^{- 2}$ .

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Step 3.5.1

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

$2 A r^{- 3} + B (- r^{3 \cdot - 2} (3 r^{2}))$

Step 3.5.2

Multiply $3$ by $- 2$ .

$2 A r^{- 3} + B (- r^{- 6} (3 r^{2}))$

Step 3.6

Multiply $3$ by $- 1$ .

$2 A r^{- 3} + B (- 3 r^{- 6} r^{2})$

Step 3.7

Multiply $r^{- 6}$ by $r^{2}$ by adding the exponents.

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Step 3.7.1

Move $r^{2}$ .

$2 A r^{- 3} + B (- 3 (r^{2} r^{- 6}))$

Step 3.7.2

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

$2 A r^{- 3} + B (- 3 r^{2 - 6})$

Step 3.7.3

Subtract $6$ from $2$ .

$2 A r^{- 3} + B (- 3 r^{- 4})$

Step 4

Simplify.

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Step 4.1

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

$2 A \frac{1}{r^{3}} + B (- 3 r^{- 4})$

Step 4.2

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

$2 A \frac{1}{r^{3}} + B (- 3 \frac{1}{r^{4}})$

Step 4.3

Combine terms.

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Step 4.3.1

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

$A \frac{2}{r^{3}} + B (- 3 \frac{1}{r^{4}})$

Step 4.3.2

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

$\frac{A \cdot 2}{r^{3}} + B (- 3 \frac{1}{r^{4}})$

Step 4.3.3

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

$\frac{2 \cdot A}{r^{3}} + B (- 3 \frac{1}{r^{4}})$

Step 4.3.4

Combine $- 3$ and $\frac{1}{r^{4}}$ .

$\frac{2 A}{r^{3}} + B \frac{- 3}{r^{4}}$

Step 4.3.5

Move the negative in front of the fraction.

$\frac{2 A}{r^{3}} + B (- \frac{3}{r^{4}})$

Step 4.3.6

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

$\frac{2 A}{r^{3}} - \frac{B \cdot 3}{r^{4}}$

Step 4.3.7

Move $3$ to the left of $B$ .

$\frac{2 A}{r^{3}} - \frac{3 B}{r^{4}}$