Evaluate the Difference Quotient f(x)=2x^2+x , a=5

$f (x) = 2 x^{2} + x$ , $a = 5$

Consider the difference quotient formula.

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

Find the components of the definition.

Evaluate the function at $x = x + h$ .

Replace the variable $x$ with $x + h$ in the expression.

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h (x + h)) + x + h$

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h x + h \cdot h) + x + h$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

Multiply $2$ by $2$ .

$f (x + h) = 2 x^{2} + 4 h x + 2 h^{2} + x + h$

The final answer is $2 x^{2} + 4 h x + 2 h^{2} + x + h$ .

$2 x^{2} + 4 h x + 2 h^{2} + x + h$

Reorder.

Move $x$ .

$2 x^{2} + 4 h x + 2 h^{2} + h + x$

Move $2 x^{2}$ .

$4 h x + 2 h^{2} + 2 x^{2} + h + x$

Reorder $4 h x$ and $2 h^{2}$ .

$2 h^{2} + 4 h x + 2 x^{2} + h + x$

Find the components of the definition.

$f (x + h) = 2 h^{2} + 4 h x + 2 x^{2} + h + x$

$f (x) = 2 x^{2} + x$

$f (x + h) = 2 h^{2} + 4 h x + 2 x^{2} + h + x$

$f (x) = 2 x^{2} + x$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 h^{2} + 4 h x + 2 x^{2} + h + x - (2 x^{2} + x)}{h}$

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{2 h^{2} + 4 h x + 2 x^{2} + h + x - (2 x^{2}) - x}{h}$

Multiply $2$ by $- 1$ .

$\frac{2 h^{2} + 4 h x + 2 x^{2} + h + x - 2 x^{2} - x}{h}$

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{2 h^{2} + 4 h x + h + x + 0 - x}{h}$

Add $2 h^{2}$ and $0$ .

$\frac{2 h^{2} + 4 h x + h + x - x}{h}$

Subtract $x$ from $x$ .

$\frac{2 h^{2} + 4 h x + h + 0}{h}$

Add $2 h^{2} + 4 h x + h$ and $0$ .

$\frac{2 h^{2} + 4 h x + h}{h}$

Factor $h$ out of $2 h^{2} + 4 h x + h$ .

Factor $h$ out of $2 h^{2}$ .

$\frac{h (2 h) + 4 h x + h}{h}$

Factor $h$ out of $4 h x$ .

$\frac{h (2 h) + h (4 x) + h}{h}$

Raise $h$ to the power of $1$ .

$\frac{h (2 h) + h (4 x) + h^{1}}{h}$

Factor $h$ out of $h^{1}$ .

$\frac{h (2 h) + h (4 x) + h \cdot 1}{h}$

Factor $h$ out of $h (2 h) + h (4 x)$ .

$\frac{h (2 h + 4 x) + h \cdot 1}{h}$

Factor $h$ out of $h (2 h + 4 x) + h \cdot 1$ .

$\frac{h (2 h + 4 x + 1)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (2 h + 4 x + 1)}{h}$

Divide $2 h + 4 x + 1$ by $1$ .

$2 h + 4 x + 1$

Reorder $2 h$ and $4 x$ .

$4 x + 2 h + 1$

Replace the variable $x$ with $5$ in the expression.

$4 (5) + 2 h + 1$

Multiply $4$ by $5$ .

$20 + 2 h + 1$

Add $20$ and $1$ .

$2 h + 21$

Evaluate the Difference Quotient f(x)=2x^2+x , a=5

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