Calculus Examples

$\frac{\sqrt{x + h} - \sqrt{x}}{h}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + h$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x + h$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x + h$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Combine $\frac{1}{2}$ and ${(x + h)}^{- \frac{1}{2}}$ .

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

Step 7.3

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

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

Step 8

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

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

Step 9

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

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

Step 10

Since $h$ is constant with respect to $x$ , the derivative of $h$ with respect to $x$ is $0$ .

$\frac{1}{h} (\frac{1}{2 {(x + h)}^{\frac{1}{2}}} (1 + 0) + \frac{d}{d x} [- x^{\frac{1}{2}}])$

Step 11

Simplify the expression.

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Step 11.1

Add $1$ and $0$ .

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

Step 11.2

Multiply $\frac{1}{2 {(x + h)}^{\frac{1}{2}}}$ by $1$ .

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Combine the numerators over the common denominator.

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

Step 17

Simplify the numerator.

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Step 17.1

Multiply $- 1$ by $2$ .

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

Step 17.2

Subtract $2$ from $1$ .

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

Step 18

Move the negative in front of the fraction.

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

Step 19

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

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

Step 20

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

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

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

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

Step 21.2

Combine terms.

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Step 21.2.1

Multiply $\frac{1}{h}$ by $\frac{1}{2 {(x + h)}^{\frac{1}{2}}}$ .

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

Step 21.2.2

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

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

Step 21.2.3

Move $2$ to the left of $h$ .

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

Step 21.3

Simplify each term.

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Step 21.3.1

Simplify the denominator.

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Step 21.3.1.1

Rewrite.

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

Step 21.3.1.2

Remove unnecessary parentheses.

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

Step 21.3.2

Move $2$ to the left of $h$ .

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