Calculus Examples

$y = \frac{a}{x} - \frac{b}{x^{2}} + \frac{c}{x^{3}}$

Step 1

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

$\frac{d}{d x} [\frac{a}{x}] + \frac{d}{d x} [- \frac{b}{x^{2}}] + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{a}{x}]$ .

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Step 2.1

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

$a \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [- \frac{b}{x^{2}}] + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$a \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [- \frac{b}{x^{2}}] + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 2.3

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

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

Step 3

Evaluate $\frac{d}{d x} [- \frac{b}{x^{2}}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

$a (- x^{- 2}) - b (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.3.2

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

$a (- x^{- 2}) - b (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

$a (- x^{- 2}) - b (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.5.2

Multiply $2$ by $- 2$ .

$a (- x^{- 2}) - b (- x^{- 4} (2 x)) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.6

Multiply $2$ by $- 1$ .

$a (- x^{- 2}) - b (- 2 x^{- 4} x) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.7

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

$a (- x^{- 2}) - b (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.8

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

$a (- x^{- 2}) - b (- 2 x^{1 - 4}) + \frac{d}{d x} [\frac{c}{x^{3}}]$

Step 3.9

Subtract $4$ from $1$ .

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

Step 3.10

Multiply $- 2$ by $- 1$ .

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

Step 4

Evaluate $\frac{d}{d x} [\frac{c}{x^{3}}]$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

$a (- x^{- 2}) + 2 b x^{- 3} + c (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{3}])$

Step 4.3.2

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

$a (- x^{- 2}) + 2 b x^{- 3} + c (- {u_{2}}^{- 2} \frac{d}{d x} [x^{3}])$

Step 4.3.3

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

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

Step 4.4

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

$a (- x^{- 2}) + 2 b x^{- 3} + c (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 4.5

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

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Step 4.5.1

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

$a (- x^{- 2}) + 2 b x^{- 3} + c (- x^{3 \cdot - 2} (3 x^{2}))$

Step 4.5.2

Multiply $3$ by $- 2$ .

$a (- x^{- 2}) + 2 b x^{- 3} + c (- x^{- 6} (3 x^{2}))$

Step 4.6

Multiply $3$ by $- 1$ .

$a (- x^{- 2}) + 2 b x^{- 3} + c (- 3 x^{- 6} x^{2})$

Step 4.7

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

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Step 4.7.1

Move $x^{2}$ .

$a (- x^{- 2}) + 2 b x^{- 3} + c (- 3 (x^{2} x^{- 6}))$

Step 4.7.2

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

$a (- x^{- 2}) + 2 b x^{- 3} + c (- 3 x^{2 - 6})$

Step 4.7.3

Subtract $6$ from $2$ .

$a (- x^{- 2}) + 2 b x^{- 3} + c (- 3 x^{- 4})$

Step 5

Simplify.

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Step 5.1

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

$a (- \frac{1}{x^{2}}) + 2 b x^{- 3} + c (- 3 x^{- 4})$

Step 5.2

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

$a (- \frac{1}{x^{2}}) + 2 b \frac{1}{x^{3}} + c (- 3 x^{- 4})$

Step 5.3

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

$a (- \frac{1}{x^{2}}) + 2 b \frac{1}{x^{3}} + c (- 3 \frac{1}{x^{4}})$

Step 5.4

Combine terms.

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Step 5.4.1

Combine $a$ and $\frac{1}{x^{2}}$ .

$- \frac{a}{x^{2}} + 2 b \frac{1}{x^{3}} + c (- 3 \frac{1}{x^{4}})$

Step 5.4.2

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

$- \frac{a}{x^{2}} + b \frac{2}{x^{3}} + c (- 3 \frac{1}{x^{4}})$

Step 5.4.3

Combine $b$ and $\frac{2}{x^{3}}$ .

$- \frac{a}{x^{2}} + \frac{b \cdot 2}{x^{3}} + c (- 3 \frac{1}{x^{4}})$

Step 5.4.4

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

$- \frac{a}{x^{2}} + \frac{2 \cdot b}{x^{3}} + c (- 3 \frac{1}{x^{4}})$

Step 5.4.5

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

$- \frac{a}{x^{2}} + \frac{2 b}{x^{3}} + c \frac{- 3}{x^{4}}$

Step 5.4.6

Move the negative in front of the fraction.

$- \frac{a}{x^{2}} + \frac{2 b}{x^{3}} + c (- \frac{3}{x^{4}})$

Step 5.4.7

Combine $c$ and $\frac{3}{x^{4}}$ .

$- \frac{a}{x^{2}} + \frac{2 b}{x^{3}} - \frac{c \cdot 3}{x^{4}}$

Step 5.4.8

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

$- \frac{a}{x^{2}} + \frac{2 b}{x^{3}} - \frac{3 c}{x^{4}}$