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

Simplify each term.

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Step 1.3.1

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

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

Step 1.3.2

Combine and simplify the denominator.

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

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

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

Step 1.3.2.4

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

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

Step 1.3.2.5

Add $1$ and $1$ .

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

Step 1.3.2.6

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

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Step 1.3.2.6.1

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

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

Step 1.3.2.6.2

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

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

Step 1.3.2.6.3

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

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

Step 1.3.2.6.4

Cancel the common factor of $2$ .

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Step 1.3.2.6.4.1

Cancel the common factor.

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

Step 1.3.2.6.4.2

Rewrite the expression.

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

Step 1.3.2.6.5

Simplify.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Write each expression with a common denominator of $(x + h) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

Reorder the factors of $(x + h) x$ .

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

Step 1.7

Combine the numerators over the common denominator.

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

Step 1.8

Simplify the numerator.

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

Apply the distributive property.

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

Step 1.8.4

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

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Step 1.8.4.1

Move $x$ .

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

Step 1.8.4.2

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

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Step 1.8.4.2.1

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

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

Step 1.8.4.2.2

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

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

Step 1.8.4.3

Write $1$ as a fraction with a common denominator.

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

Step 1.8.4.4

Combine the numerators over the common denominator.

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

Step 1.8.4.5

Add $2$ and $1$ .

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

Step 1.8.5

Rewrite ${(x + h)}^{\frac{1}{2}} x - x^{\frac{3}{2}} - x^{\frac{1}{2}} h$ in a factored form.

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Step 1.8.5.1

Factor $x^{\frac{1}{2}}$ out of ${(x + h)}^{\frac{1}{2}} x - x^{\frac{3}{2}} - x^{\frac{1}{2}} h$ .

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Step 1.8.5.1.1

Reorder the expression.

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Step 1.8.5.1.1.1

Reorder $x$ and $h$ .

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

Step 1.8.5.1.1.2

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

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

Step 1.8.5.1.1.3

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

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

Step 1.8.5.1.2

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

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

Step 1.8.5.1.3

Factor $x^{\frac{1}{2}}$ out of $- x^{\frac{3}{2}}$ .

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

Step 1.8.5.1.4

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

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

Step 1.8.5.1.5

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

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

Step 1.8.5.1.6

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

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

Step 1.8.5.2

Divide $2$ by $2$ .

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

Step 1.8.5.3

Simplify.

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

Step 1.9

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

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

Step 1.10

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

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Step 1.10.1

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

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

Step 1.10.2

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

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Step 1.10.2.1

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

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

Step 1.10.2.2

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

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

Step 1.10.3

Write $1$ as a fraction with a common denominator.

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

Step 1.10.4

Combine the numerators over the common denominator.

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

Step 1.10.5

Add $- 1$ and $2$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

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

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