Algebra Examples

$\frac{\frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}}}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\frac{1}{\sqrt{x + h}} - \frac{\sqrt{x}}{\sqrt{x} \sqrt{x}}}{h}$

Step 1.2.2

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{\frac{1}{\sqrt{x + h}} - \frac{\sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}}{h}$

Step 1.2.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{\frac{1}{\sqrt{x + h}} - \frac{\sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}}{h}$

Step 1.2.4

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

$\frac{\frac{1}{\sqrt{x + h}} - \frac{\sqrt{x}}{{\sqrt{x}}^{1 + 1}}}{h}$

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

$\frac{\frac{1}{\sqrt{x + h}} - \frac{\sqrt{x}}{x^{1}}}{h}$

Step 1.2.6.5

Simplify.

$\frac{\frac{1}{\sqrt{x + h}} - \frac{\sqrt{x}}{x}}{h}$

Step 1.3

To write $\frac{1}{\sqrt{x + h}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\frac{1}{\sqrt{x + h}} \cdot \frac{x}{x} - \frac{\sqrt{x}}{x}}{h}$

Step 1.4

To write $- \frac{\sqrt{x}}{x}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x + h}}{\sqrt{x + h}}$ .

$\frac{\frac{1}{\sqrt{x + h}} \cdot \frac{x}{x} - \frac{\sqrt{x}}{x} \cdot \frac{\sqrt{x + h}}{\sqrt{x + h}}}{h}$

Step 1.5

Write each expression with a common denominator of $\sqrt{x + h} x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{1}{\sqrt{x + h}}$ by $\frac{x}{x}$ .

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

Step 1.5.2

Multiply $\frac{\sqrt{x}}{x}$ by $\frac{\sqrt{x + h}}{\sqrt{x + h}}$ .

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

Step 1.5.3

Reorder the factors of $\sqrt{x + h} x$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Rewrite $\frac{x - \sqrt{x} \sqrt{x + h}}{x \sqrt{x + h}}$ in a factored form.

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Factor $x^{\frac{1}{2}}$ out of $x - x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}}$ .

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Step 1.7.3.1

Reorder $x$ and $h$ .

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

Step 1.7.3.2

Raise $x$ to the power of $1$ .

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

Step 1.7.3.3

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

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

Step 1.7.3.4

Factor $x^{\frac{1}{2}}$ out of $- x^{\frac{1}{2}} {(h + x)}^{\frac{1}{2}}$ .

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

Step 1.7.3.5

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} (- {(h + x)}^{\frac{1}{2}})$ .

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

Step 1.7.4

Reduce the expression $\frac{x^{\frac{1}{2}} (x^{\frac{1}{2}} - {(h + x)}^{\frac{1}{2}})}{x \sqrt{x + h}}$ by cancelling the common factors.

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Step 1.7.4.1

Factor $x$ out of $x^{\frac{1}{2}} (x^{\frac{1}{2}} - {(h + x)}^{\frac{1}{2}})$ .

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

Step 1.7.4.2

Factor $x$ out of $x \sqrt{x + h}$ .

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

Step 1.7.4.3

Cancel the common factor.

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

Step 1.7.4.4

Rewrite the expression.

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Combine and simplify the denominator.

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Step 1.10.1

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

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

Step 1.10.2

Move $\sqrt{x + h}$ .

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

Step 1.10.3

Raise $\sqrt{x + h}$ to the power of $1$ .

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

Step 1.10.4

Raise $\sqrt{x + h}$ to the power of $1$ .

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

Step 1.10.5

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

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

Step 1.10.6

Add $1$ and $1$ .

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

Step 1.10.7

Rewrite ${\sqrt{x + h}}^{2}$ as $x + h$ .

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Step 1.10.7.1

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

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

Step 1.10.7.2

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

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

Step 1.10.7.3

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

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

Step 1.10.7.4

Cancel the common factor of $2$ .

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Step 1.10.7.4.1

Cancel the common factor.

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

Step 1.10.7.4.2

Rewrite the expression.

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

Step 1.10.7.5

Simplify.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine.

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

Step 4

Simplify the expression.

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Step 4.1

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

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

Step 4.2

Reorder factors in $\frac{(x^{\frac{1}{2}} - {(h + x)}^{\frac{1}{2}}) \sqrt{x + h}}{x^{\frac{1}{2}} (x + h) h}$ .

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