Calculus Examples

$f (x) = \sqrt{x} + \frac{5}{\sqrt{x}}$

Step 1

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

$\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [\frac{5}{\sqrt{x}}]$

Step 2

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

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Step 2.1

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

$\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{5}{\sqrt{x}}]$

Step 2.2

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

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

Step 2.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [\frac{5}{\sqrt{x}}]$

Step 2.4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [\frac{5}{\sqrt{x}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [\frac{5}{\sqrt{x}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{\frac{1}{2}}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

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

Step 3.3

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

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

Step 3.4

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 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 3.6

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

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Step 3.6.1

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 3.6.2

Cancel the common factor of $2$ .

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Step 3.6.2.1

Factor $2$ out of $- 2$ .

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 3.6.2.2

Cancel the common factor.

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 3.6.2.3

Rewrite the expression.

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 3.7

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 3.8

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 3.9

Combine the numerators over the common denominator.

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 3.10.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 3.11

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 3.12

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 3.13

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 3.14

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

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Step 3.14.1

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 3.14.2

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 3.14.3

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

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 3.14.4

Combine the numerators over the common denominator.

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 3.14.5

Simplify the numerator.

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Step 3.14.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{- \frac{1}{2}} + 5 (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 3.14.5.2

Subtract $2$ from $- 1$ .

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

Step 3.14.6

Move the negative in front of the fraction.

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

Step 3.15

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

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

Step 3.16

Multiply $- 1$ by $5$ .

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

Step 3.17

Combine $- 5$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

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

Step 3.18

Move the negative in front of the fraction.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

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