Calculus Examples

$f (x) = - 2 x + 6 x$

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.

$f (x + h) = - 2 (x + h) + 6 (x + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Apply the distributive property.

$f (x + h) = - 2 x - 2 h + 6 (x + h)$

Step 2.1.2.1.2

Apply the distributive property.

$f (x + h) = - 2 x - 2 h + 6 x + 6 h$

$f (x + h) = - 2 x - 2 h + 6 x + 6 h$

Step 2.1.2.2

Simplify by adding terms.

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Step 2.1.2.2.1

Add $- 2 x$ and $6 x$ .

$f (x + h) = - 2 h + 4 x + 6 h$

Step 2.1.2.2.2

Add $- 2 h$ and $6 h$ .

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

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

Step 2.1.2.3

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

$4 h + 4 x$

$4 h + 4 x$

$4 h + 4 x$

Step 2.2

Find the components of the definition.

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

$f (x) = 4 x$

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

$f (x) = 4 x$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $4$ .

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

Step 4.1.2

Subtract $4 x$ from $4 x$ .

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

Step 4.1.3

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

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