Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

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

Step 2.1.2.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

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

Step 2.1.2.2.2

Apply the distributive property.

$f (x + h) = \sqrt{x \cdot x + x h + h (x + h) + 1}$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = \sqrt{x \cdot x + x h + h x + h \cdot h + 1}$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.4

The final answer is $\sqrt{x^{2} + 2 h x + h^{2} + 1}$ .

$\sqrt{x^{2} + 2 h x + h^{2} + 1}$

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Multiply $- 1$ by $\sqrt{x^{2} + 1}$ .

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

Step 5