Calculus Examples

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

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

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

$\sqrt{x + h - 5}$

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\sqrt{x + h - 5} - (\sqrt{x - 5})}{h}$

Step 4

Multiply $- 1$ by $\sqrt{x - 5}$ .

$f' (x) = lim_{h \to 0} \frac{\sqrt{x + h - 5} - \sqrt{x - 5}}{h}$

Step 5

Since the values of $\frac{\sqrt{x + h - 5} - \sqrt{x - 5}}{h}$ do not exist on either side of $0$ , the limit does not exist.

$Does not exist$

Step 6