Calculus Examples

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

Step 1

Expand $(\sqrt{x + h} - \sqrt{x}) (\sqrt{x + h} + \sqrt{x})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{x + h} \sqrt{x + h} + \sqrt{x + h} \sqrt{x} - \sqrt{x} \sqrt{x + h} - \sqrt{x} \sqrt{x}$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{x + h} \sqrt{x}$ and $- \sqrt{x} \sqrt{x + h}$ .

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

Step 2.1.2

Subtract $\sqrt{x} \sqrt{x + h}$ from $\sqrt{x} \sqrt{x + h}$ .

$\sqrt{x + h} \sqrt{x + h} + 0 - \sqrt{x} \sqrt{x}$

Step 2.1.3

Add $\sqrt{x + h} \sqrt{x + h}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

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

Step 2.2.2.5

Simplify.

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

Step 2.2.3

Multiply $- \sqrt{x} \sqrt{x}$ .

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Add $1$ and $1$ .

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

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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

Step 2.2.4.4

Cancel the common factor of $2$ .

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Step 2.2.4.4.1

Cancel the common factor.

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

Step 2.2.4.4.2

Rewrite the expression.

$x + h - x^{1}$

Step 2.2.4.5

Simplify.

$x + h - x$

Step 2.3

Combine the opposite terms in $x + h - x$ .

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Step 2.3.1

Subtract $x$ from $x$ .

$h + 0$

Step 2.3.2

Add $h$ and $0$ .

$h$