Calculus Examples

$f (x) = 4 x + 9$

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.

$f (x + h) = 4 (x + h) + 9$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 4 x + 4 h + 9$

Step 2.1.2.2

The final answer is $4 x + 4 h + 9$ .

$4 x + 4 h + 9$

$4 x + 4 h + 9$

$4 x + 4 h + 9$

Step 2.2

Reorder $4 x$ and $4 h$ .

$4 h + 4 x + 9$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h + 4 x + 9$

$f (x) = 4 x + 9$

$f (x + h) = 4 h + 4 x + 9$

$f (x) = 4 x + 9$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{4 h + 4 x + 9 - (4 x + 9)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{4 h + 4 x + 9 - (4 x) - 1 \cdot 9}{h}$

Step 4.1.2

Multiply $4$ by $- 1$ .

$\frac{4 h + 4 x + 9 - 4 x - 1 \cdot 9}{h}$

Step 4.1.3

Multiply $- 1$ by $9$ .

$\frac{4 h + 4 x + 9 - 4 x - 9}{h}$

Step 4.1.4

Subtract $4 x$ from $4 x$ .

$\frac{4 h + 0 + 9 - 9}{h}$

Step 4.1.5

Add $4 h$ and $0$ .

$\frac{4 h + 9 - 9}{h}$

Step 4.1.6

Subtract $9$ from $9$ .

$\frac{4 h + 0}{h}$

Step 4.1.7

Add $4 h$ and $0$ .

$\frac{4 h}{h}$

$\frac{4 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{4 h}{h}$

Step 4.2.2

Divide $4$ by $1$ .

$4$

$4$

$4$

Step 5

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$