Calculus Examples

$f (x) = \frac{4}{\sqrt{x}} - \frac{5}{x} + \frac{8}{x^{4}}$

Step 1

By the Sum Rule, the derivative of $\frac{4}{\sqrt{x}} - \frac{5}{x} + \frac{8}{x^{4}}$ with respect to $x$ is $\frac{d}{d x} [\frac{4}{\sqrt{x}}] + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$ .

$\frac{d}{d x} [\frac{4}{\sqrt{x}}] + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2

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

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{4}{x^{\frac{1}{2}}}] + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{\frac{1}{2}}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$4 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}] + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.3

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

$4 \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}] + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.4.1

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

$4 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.4.2

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

$4 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.4.3

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

$4 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.5

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

$4 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 2.6.1

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

$4 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.6.2

Cancel the common factor of $2$ .

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Step 2.6.2.1

Factor $2$ out of $- 2$ .

$4 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.6.2.2

Cancel the common factor.

$4 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.6.2.3

Rewrite the expression.

$4 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.8

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

$4 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.9

Combine the numerators over the common denominator.

$4 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$4 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.10.2

Subtract $2$ from $1$ .

$4 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.11

Move the negative in front of the fraction.

$4 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.12

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

$4 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.13

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

$4 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 2.14.1

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

$4 (- \frac{x^{- \frac{1}{2} - 1}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14.3

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

$4 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14.4

Combine the numerators over the common denominator.

$4 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14.5

Simplify the numerator.

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Step 2.14.5.1

Multiply $- 1$ by $2$ .

$4 (- \frac{x^{\frac{- 1 - 2}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14.5.2

Subtract $2$ from $- 1$ .

$4 (- \frac{x^{\frac{- 3}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.14.6

Move the negative in front of the fraction.

$4 (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.15

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{2 x^{\frac{3}{2}}}) + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.16

Multiply $- 1$ by $4$ .

$- 4 \frac{1}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.17

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

$\frac{- 4}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.18

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.19

Cancel the common factors.

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Step 2.19.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$\frac{2 \cdot - 2}{2 (x^{\frac{3}{2}})} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.19.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.19.3

Rewrite the expression.

$\frac{- 2}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 2.20

Move the negative in front of the fraction.

$- \frac{2}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{5}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{5}{x}]$ .

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Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x}]$ .

$- \frac{2}{x^{\frac{3}{2}}} - 5 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 3.2

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

$- \frac{2}{x^{\frac{3}{2}}} - 5 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 3.3

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

$- \frac{2}{x^{\frac{3}{2}}} - 5 (- x^{- 2}) + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 3.4

Multiply $- 1$ by $- 5$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + \frac{d}{d x} [\frac{8}{x^{4}}]$

Step 4

Evaluate $\frac{d}{d x} [\frac{8}{x^{4}}]$ .

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Step 4.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x^{4}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 4.2

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 \frac{d}{d x} [{(x^{4})}^{- 1}]$

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

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Step 4.3.1

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{4}])$

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 $n = - 1$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{4}])$

Step 4.3.3

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 4.4

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 4.5

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

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Step 4.5.1

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- x^{4 \cdot - 2} (4 x^{3}))$

Step 4.5.2

Multiply $4$ by $- 2$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- x^{- 8} (4 x^{3}))$

Step 4.6

Multiply $4$ by $- 1$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- 4 x^{- 8} x^{3})$

Step 4.7

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

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Step 4.7.1

Move $x^{3}$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- 4 (x^{3} x^{- 8}))$

Step 4.7.2

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- 4 x^{3 - 8})$

Step 4.7.3

Subtract $8$ from $3$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} + 8 (- 4 x^{- 5})$

Step 4.8

Multiply $- 4$ by $8$ .

$- \frac{2}{x^{\frac{3}{2}}} + 5 x^{- 2} - 32 x^{- 5}$

Step 5

Simplify.

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Step 5.1

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 \frac{1}{x^{2}} - 32 x^{- 5}$

Step 5.2

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

$- \frac{2}{x^{\frac{3}{2}}} + 5 \frac{1}{x^{2}} - 32 \frac{1}{x^{5}}$

Step 5.3

Combine terms.

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Step 5.3.1

Combine $5$ and $\frac{1}{x^{2}}$ .

$- \frac{2}{x^{\frac{3}{2}}} + \frac{5}{x^{2}} - 32 \frac{1}{x^{5}}$

Step 5.3.2

Combine $- 32$ and $\frac{1}{x^{5}}$ .

$- \frac{2}{x^{\frac{3}{2}}} + \frac{5}{x^{2}} + \frac{- 32}{x^{5}}$

Step 5.3.3

Move the negative in front of the fraction.

$- \frac{2}{x^{\frac{3}{2}}} + \frac{5}{x^{2}} - \frac{32}{x^{5}}$

Step 5.4

Reorder terms.

$\frac{5}{x^{2}} - \frac{32}{x^{5}} - \frac{2}{x^{\frac{3}{2}}}$