Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{2}{x} - \frac{2}{x^{3}} + \frac{1}{x^{4}}$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{x}] + \frac{d}{d x} [- \frac{2}{x^{3}}] + \frac{d}{d x} [\frac{1}{x^{4}}]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $2$ .

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

Step 3

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

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{3}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

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

Step 3.2

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

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

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 $g (x) = x^{3}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

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

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 $n = - 1$ .

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

Step 3.3.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Multiply $3$ by $- 2$ .

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

Step 3.6

Multiply $3$ by $- 1$ .

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

Step 3.7

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

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Step 3.7.1

Move $x^{2}$ .

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

Step 3.7.2

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

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

Step 3.7.3

Subtract $6$ from $2$ .

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

Step 3.8

Multiply $- 3$ by $- 2$ .

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

Step 4

Evaluate $\frac{d}{d x} [\frac{1}{x^{4}}]$ .

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Step 4.1

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

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

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 $g (x) = x^{4}$ .

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Step 4.2.1

To apply the Chain Rule, set $u_{2}$ as $x^{4}$ .

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

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 $n = - 1$ .

$- 2 x^{- 2} + 6 x^{- 4} - {u_{2}}^{- 2} \frac{d}{d x} [x^{4}]$

Step 4.2.3

Replace all occurrences of $u_{2}$ with $x^{4}$ .

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

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 2 x^{- 2} + 6 x^{- 4} - {(x^{4})}^{- 2} (4 x^{3})$

Step 4.4

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 4.4.1

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

$- 2 x^{- 2} + 6 x^{- 4} - x^{4 \cdot - 2} (4 x^{3})$

Step 4.4.2

Multiply $4$ by $- 2$ .

$- 2 x^{- 2} + 6 x^{- 4} - x^{- 8} (4 x^{3})$

Step 4.5

Multiply $4$ by $- 1$ .

$- 2 x^{- 2} + 6 x^{- 4} - 4 x^{- 8} x^{3}$

Step 4.6

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 4.6.1

Move $x^{3}$ .

$- 2 x^{- 2} + 6 x^{- 4} - 4 (x^{3} x^{- 8})$

Step 4.6.2

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

$- 2 x^{- 2} + 6 x^{- 4} - 4 x^{3 - 8}$

Step 4.6.3

Subtract $8$ from $3$ .

$- 2 x^{- 2} + 6 x^{- 4} - 4 x^{- 5}$

Step 5

Simplify.

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Step 5.1

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

$- 2 \frac{1}{x^{2}} + 6 x^{- 4} - 4 x^{- 5}$

Step 5.2

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

$- 2 \frac{1}{x^{2}} + 6 \frac{1}{x^{4}} - 4 x^{- 5}$

Step 5.3

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

$- 2 \frac{1}{x^{2}} + 6 \frac{1}{x^{4}} - 4 \frac{1}{x^{5}}$

Step 5.4

Combine terms.

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Step 5.4.1

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

$\frac{- 2}{x^{2}} + 6 \frac{1}{x^{4}} - 4 \frac{1}{x^{5}}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{2}{x^{2}} + 6 \frac{1}{x^{4}} - 4 \frac{1}{x^{5}}$

Step 5.4.3

Combine $6$ and $\frac{1}{x^{4}}$ .

$- \frac{2}{x^{2}} + \frac{6}{x^{4}} - 4 \frac{1}{x^{5}}$

Step 5.4.4

Combine $- 4$ and $\frac{1}{x^{5}}$ .

$- \frac{2}{x^{2}} + \frac{6}{x^{4}} + \frac{- 4}{x^{5}}$

Step 5.4.5

Move the negative in front of the fraction.

$- \frac{2}{x^{2}} + \frac{6}{x^{4}} - \frac{4}{x^{5}}$