Calculus Examples

$g (x) = 3 \sqrt{x} + 2$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

The final answer is $3 \sqrt{x + h} + 2$ .

$3 \sqrt{x + h} + 2$

Step 2.2

Find the components of the definition.

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

$g (x) = 3 \sqrt{x} + 2$

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

$g (x) = 3 \sqrt{x} + 2$

Step 3

Plug in the components.

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

Step 4

Simplify the numerator.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + h}$ as ${(x + h)}^{\frac{1}{2}}$ .

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

Step 4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4.3

Let $u = {(x + h)}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${(x + h)}^{\frac{1}{2}}$ .

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Step 4.3.1

Subtract $2$ from $2$ .

$\frac{3 u - 3 x^{\frac{1}{2}} + 0}{h}$

Step 4.3.2

Add $3 u - 3 x^{\frac{1}{2}}$ and $0$ .

$\frac{3 u - 3 x^{\frac{1}{2}}}{h}$

Step 4.4

Factor $3$ out of $3 u - 3 x^{\frac{1}{2}}$ .

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Step 4.4.1

Factor $3$ out of $3 u$ .

$\frac{3 (u) - 3 x^{\frac{1}{2}}}{h}$

Step 4.4.2

Factor $3$ out of $- 3 x^{\frac{1}{2}}$ .

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

Step 4.4.3

Factor $3$ out of $3 (u) + 3 (- x^{\frac{1}{2}})$ .

$\frac{3 (u - x^{\frac{1}{2}})}{h}$

Step 4.5

Replace all occurrences of $u$ with ${(x + h)}^{\frac{1}{2}}$ .

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

Step 5