Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{- 4}{x^{3}} + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [\frac{- 4}{x^{3}}] + \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{- 4}{x^{3}}]$ .

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4}{x^{3}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

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

Step 2.3

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

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

Step 2.4

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 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Multiply $3$ by $- 2$ .

$- 4 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [\frac{1}{x}]$

Step 2.7

Multiply $3$ by $- 1$ .

$- 4 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [\frac{1}{x}]$

Step 2.8

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

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Step 2.8.1

Move $x^{2}$ .

$- 4 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [\frac{1}{x}]$

Step 2.8.2

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

$- 4 (- 3 x^{2 - 6}) + \frac{d}{d x} [\frac{1}{x}]$

Step 2.8.3

Subtract $6$ from $2$ .

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

Step 2.9

Multiply $- 3$ by $- 4$ .

$12 x^{- 4} + \frac{d}{d x} [\frac{1}{x}]$

Step 3

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

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Step 3.1

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

$12 x^{- 4} + \frac{d}{d x} [x^{- 1}]$

Step 3.2

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

$12 x^{- 4} - x^{- 2}$

Step 4

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

$12 \frac{1}{x^{4}} - x^{- 2}$

Step 5

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

$12 \frac{1}{x^{4}} - \frac{1}{x^{2}}$

Step 6

Simplify.

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Step 6.1

Combine $12$ and $\frac{1}{x^{4}}$ .

$\frac{12}{x^{4}} - \frac{1}{x^{2}}$

Step 6.2

Reorder terms.

$- \frac{1}{x^{2}} + \frac{12}{x^{4}}$