Calculus Examples

$f (x) = \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$\frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{x^{2}}]$ .

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Step 3.1

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

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

Step 3.2

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 3.2.1

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

$- x^{- 2} + \frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2.2

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

$- x^{- 2} - {u_{1}}^{- 2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Multiply $2$ by $- 2$ .

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

Step 3.5

Multiply $2$ by $- 1$ .

$- x^{- 2} - 2 x^{- 4} x + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.6

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

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

Step 3.7

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

$- x^{- 2} - 2 x^{1 - 4} + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.8

Subtract $4$ from $1$ .

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

Step 4

Evaluate $\frac{d}{d x} [\frac{1}{x^{3}}]$ .

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Step 4.1

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

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

Step 4.2

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 4.2.1

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

$- x^{- 2} - 2 x^{- 3} + \frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{3}]$

Step 4.2.2

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

$- x^{- 2} - 2 x^{- 3} - {u_{2}}^{- 2} \frac{d}{d x} [x^{3}]$

Step 4.2.3

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

$- x^{- 2} - 2 x^{- 3} - {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]$

Step 4.3

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

$- x^{- 2} - 2 x^{- 3} - {(x^{3})}^{- 2} (3 x^{2})$

Step 4.4

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

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Step 4.4.1

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

$- x^{- 2} - 2 x^{- 3} - x^{3 \cdot - 2} (3 x^{2})$

Step 4.4.2

Multiply $3$ by $- 2$ .

$- x^{- 2} - 2 x^{- 3} - x^{- 6} (3 x^{2})$

Step 4.5

Multiply $3$ by $- 1$ .

$- x^{- 2} - 2 x^{- 3} - 3 x^{- 6} x^{2}$

Step 4.6

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

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Step 4.6.1

Move $x^{2}$ .

$- x^{- 2} - 2 x^{- 3} - 3 (x^{2} x^{- 6})$

Step 4.6.2

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

$- x^{- 2} - 2 x^{- 3} - 3 x^{2 - 6}$

Step 4.6.3

Subtract $6$ from $2$ .

$- x^{- 2} - 2 x^{- 3} - 3 x^{- 4}$

Step 5

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

$- \frac{1}{x^{2}} - 2 x^{- 3} - 3 x^{- 4}$

Step 6

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

$- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}} - 3 x^{- 4}$

Step 7

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

$- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 8

Combine terms.

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Step 8.1

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

$- \frac{1}{x^{2}} + \frac{- 2}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{1}{x^{2}} - \frac{2}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 8.3

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

$- \frac{1}{x^{2}} - \frac{2}{x^{3}} + \frac{- 3}{x^{4}}$

Step 8.4

Move the negative in front of the fraction.

$- \frac{1}{x^{2}} - \frac{2}{x^{3}} - \frac{3}{x^{4}}$