Calculus Examples

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

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.2.2

The final answer is $\frac{1}{{(x + h)}^{\frac{1}{2}}}$ .

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1}{{(x + h)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

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

Step 4.1.2

To write $- \frac{1}{x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(x + h)}^{\frac{1}{2}}}{{(x + h)}^{\frac{1}{2}}}$ .

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

Step 4.1.3

Write each expression with a common denominator of ${(x + h)}^{\frac{1}{2}} x^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

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

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

Step 4.4

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

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

Step 5