Precalculus Examples

$y = - \sqrt{2 x + 5}$

Step 1

Write $y = - \sqrt{2 x + 5}$ as a function.

$f (x) = - \sqrt{2 x + 5}$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = - \sqrt{2 (x + h) + 5}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = - \sqrt{2 x + 2 h + 5}$

Step 3.1.2.2

The final answer is $- \sqrt{2 x + 2 h + 5}$ .

$- \sqrt{2 x + 2 h + 5}$

Step 3.2

Find the components of the definition.

$f (x + h) = - \sqrt{2 x + 2 h + 5}$

$f (x) = - \sqrt{2 x + 5}$

$f (x + h) = - \sqrt{2 x + 2 h + 5}$

$f (x) = - \sqrt{2 x + 5}$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \sqrt{2 x + 2 h + 5} - (- \sqrt{2 x + 5})}{h}$

Step 5

Simplify.

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Step 5.1

Factor $- 1$ out of $- \sqrt{2 x + 2 h + 5} - - \sqrt{2 x + 5}$ .

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Step 5.1.1

Move parentheses.

$\frac{- \sqrt{2 x + 2 h + 5} - - 1 \sqrt{2 x + 5}}{h}$

Step 5.1.2

Factor $- 1$ out of $- \sqrt{2 x + 2 h + 5}$ .

$\frac{- \sqrt{2 x + 2 h + 5} - - 1 \sqrt{2 x + 5}}{h}$

Step 5.1.3

Factor $- 1$ out of $- - 1 \sqrt{2 x + 5}$ .

$\frac{- \sqrt{2 x + 2 h + 5} - - \sqrt{2 x + 5}}{h}$

Step 5.1.4

Factor $- 1$ out of $- \sqrt{2 x + 2 h + 5} - - \sqrt{2 x + 5}$ .

$\frac{- (\sqrt{2 x + 2 h + 5} - \sqrt{2 x + 5})}{h}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{\sqrt{2 x + 2 h + 5} - \sqrt{2 x + 5}}{h}$

Step 6