Calculus Examples

f (x) = \sqrt{x} - \frac{1}{\sqrt{x}}

$f (x) = \sqrt{x} - \frac{1}{\sqrt{x}}$

Step 1

This derivative could not be completed using the quotient rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of

\sqrt{x} - \frac{1}{\sqrt{x}}

$\sqrt{x} - \frac{1}{\sqrt{x}}$ with respect to

x

$x$ is

\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]

$\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]$ .

\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- \frac{1}{\sqrt{x}}]

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

Step 3

Evaluate

\frac{d}{d x} [\sqrt{x}]

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

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

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

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

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

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

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

Step 3.3

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 3.4

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 3.5

Combine the numerators over the common denominator.

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

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 3.6.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 3.7

Move the negative in front of the fraction.

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

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

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

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

Step 4

Evaluate

\frac{d}{d x} [- \frac{1}{\sqrt{x}}]

$\frac{d}{d x} [- \frac{1}{\sqrt{x}}]$ .

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Step 4.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

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

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

Step 4.2

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = - 1

$f (x) = - 1$ and

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

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

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

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

Step 4.3

Rewrite

\frac{1}{x^{\frac{1}{2}}}

$\frac{1}{x^{\frac{1}{2}}}$ as

{(x^{\frac{1}{2}})}^{- 1}

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

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

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

Step 4.4

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{- 1}

$f (x) = x^{- 1}$ and

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

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

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Step 4.4.1

To apply the Chain Rule, set

u

$u$ as

x^{\frac{1}{2}}

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

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

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

Step 4.4.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = - 1

$n = - 1$ .

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

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

Step 4.4.3

Replace all occurrences of

u

$u$ with

x^{\frac{1}{2}}

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

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

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

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

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

Step 4.5

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

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

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

Step 4.6

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

\frac{1}{2} x^{- \frac{1}{2}} - (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0

$\frac{1}{2} x^{- \frac{1}{2}} - (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7

Multiply the exponents in

{(x^{\frac{1}{2}})}^{- 2}

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

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Step 4.7.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{1}{2} x^{- \frac{1}{2}} - (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7.2

Cancel the common factor of

2

$2$ .

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Step 4.7.2.1

Factor

2

$2$ out of

- 2

$- 2$ .

\frac{1}{2} x^{- \frac{1}{2}} - (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7.2.2

Cancel the common factor.

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.7.2.3

Rewrite the expression.

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.8

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.9

Combine

$- 1$ and

$\frac{2}{2}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.10

Combine the numerators over the common denominator.

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Multiply

$- 1$ by

$2$ .

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.11.2

Subtract

$2$ from

$1$ .

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.12

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.13

Combine

$\frac{1}{2}$ and

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

$\frac{1}{2} x^{- \frac{1}{2}} - (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.14

Combine

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

$x^{- 1}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{- \frac{1}{2}} x^{- 1}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15

Multiply

$x^{- \frac{1}{2}}$ by

$x^{- 1}$ by adding the exponents.

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Step 4.15.1

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{- \frac{1}{2} - 1}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.2

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.3

Combine

$- 1$ and

$\frac{2}{2}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.4

Combine the numerators over the common denominator.

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.5

Simplify the numerator.

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Step 4.15.5.1

Multiply

$- 1$ by

$2$ .

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{\frac{- 1 - 2}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.5.2

Subtract

$2$ from

$- 1$ .

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{\frac{- 3}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.15.6

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{x^{- \frac{3}{2}}}{2} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.16

Move

$x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - - \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.17

Multiply

$- 1$ by

$- 1$ .

$\frac{1}{2} x^{- \frac{1}{2}} + 1 \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.18

Multiply

$\frac{1}{2 x^{\frac{3}{2}}}$ by

$1$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{1}{2}}} \cdot 0$

Step 4.19

Multiply

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

$0$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2 x^{\frac{3}{2}}} + 0$

Step 4.20

Add

$\frac{1}{2 x^{\frac{3}{2}}}$ and

$0$ .

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

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + \frac{1}{2 x^{\frac{3}{2}}}$

Step 5.2

Multiply

$\frac{1}{2}$ by

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

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