Algebra Examples

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

Simplify the numerator.

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Simplify.

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Combine using the product rule for radicals.

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

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

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

Combine and simplify the denominator.

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Multiply $\frac{x - \sqrt{(x + h) x}}{x \sqrt{x + h}}$ and $\frac{\sqrt{x + h}}{\sqrt{x + h}}$ .

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

Move $\sqrt{x + h}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Simplify.

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

Apply the distributive property.

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

Multiply $- \sqrt{(x + h) x} \sqrt{x + h}$ .

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Combine using the product rule for radicals.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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Pull terms out from under the radical.

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

Apply the distributive property.

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

Apply the distributive property.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{x \sqrt{x + h} - x \sqrt{x} - h \sqrt{x}}{x (x + h)}$ and $\frac{1}{h}$ .

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

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

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