Calculus Examples

$f (x) = \sqrt{6 x - 3}$

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.

$f (x + h) = \sqrt{6 (x + h) - 3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Factor $3$ out of $6 (x + h) - 3$ .

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Step 2.1.2.1.1

Factor $3$ out of $6 (x + h)$ .

$f (x + h) = \sqrt{3 (2 (x + h)) - 3}$

Step 2.1.2.1.2

Factor $3$ out of $- 3$ .

$f (x + h) = \sqrt{3 (2 (x + h)) + 3 (- 1)}$

Step 2.1.2.1.3

Factor $3$ out of $3 (2 (x + h)) + 3 (- 1)$ .

$f (x + h) = \sqrt{3 (2 (x + h) - 1)}$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = \sqrt{3 (2 x + 2 h - 1)}$

Step 2.1.2.3

The final answer is $\sqrt{3 (2 x + 2 h - 1)}$ .

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

Step 2.2

Find the components of the definition.

$f (x + h) = \sqrt{3 (2 x + 2 h - 1)}$

$f (x) = \sqrt{3 (2 x - 1)}$

$f (x + h) = \sqrt{3 (2 x + 2 h - 1)}$

$f (x) = \sqrt{3 (2 x - 1)}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\sqrt{3 (2 x + 2 h - 1)} - (\sqrt{3 (2 x - 1)})}{h}$

Step 4

Multiply $- 1$ by $\sqrt{3 (2 x - 1)}$ .

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

Step 5