Calculus Examples

$f (x) = \frac{7}{\sqrt{x}}$

Step 1

Multiply $\frac{7}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (x) = \frac{7}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{7}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (x) = \frac{7 \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.2

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

$f (x) = \frac{7 \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.3

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

$f (x) = \frac{7 \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.4

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

$f (x) = \frac{7 \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$f (x) = \frac{7 \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$f (x) = \frac{7 \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$f (x) = \frac{7 \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$f (x) = \frac{7 \sqrt{x}}{x^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$f (x) = \frac{7 \sqrt{x}}{x^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$f (x) = \frac{7 \sqrt{x}}{x}$

Step 2.6.5

Simplify.

$f (x) = \frac{7 \sqrt{x}}{x}$

Step 3

Consider the difference quotient formula.

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

Step 4

Find the components of the definition.

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Step 4.1

Evaluate the function at $x = x + h$ .

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Step 4.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{7 \sqrt{x + h}}{x + h}$

Step 4.1.2

The final answer is $\frac{7 \sqrt{x + h}}{x + h}$ .

$\frac{7 \sqrt{x + h}}{x + h}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{7 \sqrt{x + h}}{x + h}$

$f (x) = \frac{7 \sqrt{x}}{x}$

$f (x + h) = \frac{7 \sqrt{x + h}}{x + h}$

$f (x) = \frac{7 \sqrt{x}}{x}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{7 \sqrt{x + h}}{x + h} - (\frac{7 \sqrt{x}}{x})}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

To write $\frac{7 \sqrt{x + h}}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\frac{7 \sqrt{x + h}}{x + h} \cdot \frac{x}{x} - \frac{7 \sqrt{x}}{x}}{h}$

Step 6.1.2

To write $- \frac{7 \sqrt{x}}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

$\frac{\frac{7 \sqrt{x + h}}{x + h} \cdot \frac{x}{x} - \frac{7 \sqrt{x}}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 6.1.3

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 6.1.3.1

Multiply $\frac{7 \sqrt{x + h}}{x + h}$ by $\frac{x}{x}$ .

$\frac{\frac{7 \sqrt{x + h} x}{(x + h) x} - \frac{7 \sqrt{x}}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 6.1.3.2

Multiply $\frac{7 \sqrt{x}}{x}$ by $\frac{x + h}{x + h}$ .

$\frac{\frac{7 \sqrt{x + h} x}{(x + h) x} - \frac{7 \sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.3.3

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

$\frac{\frac{7 \sqrt{x + h} x}{x (x + h)} - \frac{7 \sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.4

Combine the numerators over the common denominator.

$\frac{\frac{7 \sqrt{x + h} x - 7 \sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.5

Rewrite $\frac{7 \sqrt{x + h} x - 7 \sqrt{x} (x + h)}{x (x + h)}$ in a factored form.

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Step 6.1.5.1

Factor $7$ out of $7 \sqrt{x + h} x - 7 \sqrt{x} (x + h)$ .

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Step 6.1.5.1.1

Factor $7$ out of $7 \sqrt{x + h} x$ .

$\frac{\frac{7 (\sqrt{x + h} x) - 7 \sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.5.1.2

Factor $7$ out of $- 7 \sqrt{x} (x + h)$ .

$\frac{\frac{7 (\sqrt{x + h} x) + 7 (- \sqrt{x} (x + h))}{x (x + h)}}{h}$

Step 6.1.5.1.3

Factor $7$ out of $7 (\sqrt{x + h} x) + 7 (- \sqrt{x} (x + h))$ .

$\frac{\frac{7 (\sqrt{x + h} x - \sqrt{x} (x + h))}{x (x + h)}}{h}$

Step 6.1.5.2

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

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - \sqrt{x} (x + h))}{x (x + h)}}{h}$

Step 6.1.5.3

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

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - x^{\frac{1}{2}} (x + h))}{x (x + h)}}{h}$

Step 6.1.5.4

Apply the distributive property.

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - x^{\frac{1}{2}} x - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.5

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

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Step 6.1.5.5.1

Move $x$ .

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - (x \cdot x^{\frac{1}{2}}) - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.5.2

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

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Step 6.1.5.5.2.1

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

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - (x^{1} x^{\frac{1}{2}}) - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.5.2.2

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

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - x^{1 + \frac{1}{2}} - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.5.3

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

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - x^{\frac{2}{2} + \frac{1}{2}} - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.5.4

Combine the numerators over the common denominator.

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - x^{\frac{2 + 1}{2}} - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.5.5

Add $2$ and $1$ .

$\frac{\frac{7 ({(x + h)}^{\frac{1}{2}} x - x^{\frac{3}{2}} - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.6

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 6.1.5.6.1

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 6.1.5.6.1.1

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

$\frac{\frac{7 (x {(x + h)}^{\frac{1}{2}} - x^{\frac{3}{2}} - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.6.1.2

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

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}}) - x^{\frac{3}{2}} - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.6.1.3

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

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}}) + x^{\frac{1}{2}} (- x^{\frac{2}{2}}) - x^{\frac{1}{2}} h)}{x (x + h)}}{h}$

Step 6.1.5.6.1.4

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

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}}) + x^{\frac{1}{2}} (- x^{\frac{2}{2}}) + x^{\frac{1}{2}} (- h))}{x (x + h)}}{h}$

Step 6.1.5.6.1.5

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

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x^{\frac{2}{2}}) + x^{\frac{1}{2}} (- h))}{x (x + h)}}{h}$

Step 6.1.5.6.1.6

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

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x^{\frac{2}{2}} - h))}{x (x + h)}}{h}$

Step 6.1.5.6.2

Divide $2$ by $2$ .

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x^{1} - h))}{x (x + h)}}{h}$

Step 6.1.5.6.3

Simplify.

$\frac{\frac{7 (x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h))}{x (x + h)}}{h}$

$\frac{\frac{7 x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{x (x + h)}}{h}$

Step 6.1.5.7

Reduce the expression $\frac{7 x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{x (x + h)}$ by cancelling the common factors.

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Step 6.1.5.7.1

Factor $x$ out of $7 x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)$ .

$\frac{\frac{x (7 x^{- \frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h))}{x (x + h)}}{h}$

Step 6.1.5.7.2

Cancel the common factor.

$\frac{\frac{x (7 x^{- \frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h))}{x (x + h)}}{h}$

Step 6.1.5.7.3

Rewrite the expression.

$\frac{\frac{7 x^{- \frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{x + h}}{h}$

Step 6.1.6

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

$\frac{\frac{7 (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{(x + h) x^{\frac{1}{2}}}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{7 (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{(x + h) x^{\frac{1}{2}}} \cdot \frac{1}{h}$

Step 6.3

Combine.

$\frac{7 (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h) \cdot 1}{(x + h) x^{\frac{1}{2}} h}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Multiply $7$ by $1$ .

$\frac{7 (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{(x + h) x^{\frac{1}{2}} h}$

Step 6.4.2

Reorder factors in $\frac{7 (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{(x + h) x^{\frac{1}{2}} h}$ .

$\frac{7 (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}} - x - h)}{h x^{\frac{1}{2}} (x + h)}$

Step 7