Algebra Examples

$h (x) = \sqrt{57 x^{2} + 27}$

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.

$h (x + h) = \sqrt{57 {(x + h)}^{2} + 27}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify with factoring out.

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Step 2.1.2.1.1

Factor $3$ out of $57 {(x + h)}^{2} + 27$ .

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Step 2.1.2.1.1.1

Factor $3$ out of $57 {(x + h)}^{2}$ .

$h (x + h) = \sqrt{3 (19 {(x + h)}^{2}) + 27}$

Step 2.1.2.1.1.2

Factor $3$ out of $27$ .

$h (x + h) = \sqrt{3 (19 {(x + h)}^{2}) + 3 (9)}$

Step 2.1.2.1.1.3

Factor $3$ out of $3 (19 {(x + h)}^{2}) + 3 (9)$ .

$h (x + h) = \sqrt{3 (19 {(x + h)}^{2} + 9)}$

Step 2.1.2.1.2

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

$h (x + h) = \sqrt{3 (19 ((x + h) (x + h)) + 9)}$

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.

$h (x + h) = \sqrt{3 (19 (x (x + h) + h (x + h)) + 9)}$

Step 2.1.2.2.2

Apply the distributive property.

$h (x + h) = \sqrt{3 (19 (x \cdot x + x h + h (x + h)) + 9)}$

Step 2.1.2.2.3

Apply the distributive property.

$h (x + h) = \sqrt{3 (19 (x \cdot x + x h + h x + h \cdot h) + 9)}$

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 $x$ .

$h (x + h) = \sqrt{3 (19 (x^{2} + x h + h x + h \cdot h) + 9)}$

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

$h (x + h) = \sqrt{3 (19 (x^{2} + x h + h x + h^{2}) + 9)}$

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 $h$ .

$h (x + h) = \sqrt{3 (19 (x^{2} + h x + h x + h^{2}) + 9)}$

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

$h (x + h) = \sqrt{3 (19 (x^{2} + 2 h x + h^{2}) + 9)}$

Step 2.1.2.4

Apply the distributive property.

$h (x + h) = \sqrt{3 (19 x^{2} + 19 (2 h x) + 19 h^{2} + 9)}$

Step 2.1.2.5

Multiply $2$ by $19$ .

$h (x + h) = \sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)}$

Step 2.1.2.6

The final answer is $\sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)}$ .

$\sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)}$

Step 2.2

Find the components of the definition.

$h (x + h) = \sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)}$

$h (x) = \sqrt{3 (19 x^{2} + 9)}$

$h (x + h) = \sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)}$

$h (x) = \sqrt{3 (19 x^{2} + 9)}$

Step 3

Plug in the components.

$\frac{h (x + h) - h x}{h} = \frac{\sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)} - (\sqrt{3 (19 x^{2} + 9)})}{h}$

Step 4

Multiply $- 1$ by $\sqrt{3 (19 x^{2} + 9)}$ .

$\frac{\sqrt{3 (19 x^{2} + 38 h x + 19 h^{2} + 9)} - \sqrt{3 (19 x^{2} + 9)}}{h}$

Step 5