Find the Difference Quotient f(x)=4x^2+5

$f (x) = 4 x^{2} + 5$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

$f (x + h) = 4 {(x + h)}^{2} + 5$

Simplify the result.

Simplify each term.

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

$f (x + h) = 4 ((x + h) (x + h)) + 5$

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

Apply the distributive property.

$f (x + h) = 4 (x (x + h) + h (x + h)) + 5$

Apply the distributive property.

$f (x + h) = 4 (x \cdot x + x h + h (x + h)) + 5$

Apply the distributive property.

$f (x + h) = 4 (x \cdot x + x h + h x + h \cdot h) + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = 4 (x^{2} + x h + h x + h \cdot h) + 5$

Multiply $h$ by $h$ .

$f (x + h) = 4 (x^{2} + x h + h x + h^{2}) + 5$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 4 (x^{2} + h x + h x + h^{2}) + 5$

Add $h x$ and $h x$ .

$f (x + h) = 4 (x^{2} + 2 h x + h^{2}) + 5$

Apply the distributive property.

$f (x + h) = 4 x^{2} + 4 (2 h x) + 4 h^{2} + 5$

Multiply $2$ by $4$ .

$f (x + h) = 4 x^{2} + 8 h x + 4 h^{2} + 5$

The final answer is $4 x^{2} + 8 h x + 4 h^{2} + 5$ .

$4 x^{2} + 8 h x + 4 h^{2} + 5$

Reorder.

Move $4 x^{2}$ .

$8 h x + 4 h^{2} + 4 x^{2} + 5$

Reorder $8 h x$ and $4 h^{2}$ .

$4 h^{2} + 8 h x + 4 x^{2} + 5$

Find the components of the definition.

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2} + 5$

$f (x) = 4 x^{2} + 5$

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2} + 5$

$f (x) = 4 x^{2} + 5$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{4 h^{2} + 8 h x + 4 x^{2} + 5 - (4 x^{2} + 5)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{4 h^{2} + 8 h x + 4 x^{2} + 5 - (4 x^{2}) - 1 \cdot 5}{h}$

Multiply $4$ by $- 1$ .

$\frac{4 h^{2} + 8 h x + 4 x^{2} + 5 - 4 x^{2} - 1 \cdot 5}{h}$

Multiply $- 1$ by $5$ .

$\frac{4 h^{2} + 8 h x + 4 x^{2} + 5 - 4 x^{2} - 5}{h}$

Subtract $4 x^{2}$ from $4 x^{2}$ .

$\frac{4 h^{2} + 8 h x + 0 + 5 - 5}{h}$

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

$\frac{4 h^{2} + 8 h x + 5 - 5}{h}$

Subtract $5$ from $5$ .

$\frac{4 h^{2} + 8 h x + 0}{h}$

Add $4 h^{2} + 8 h x$ and $0$ .

$\frac{4 h^{2} + 8 h x}{h}$

Factor $4 h$ out of $4 h^{2} + 8 h x$ .

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

$\frac{4 h \cdot h + 8 h x}{h}$

Factor $4 h$ out of $8 h x$ .

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

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

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

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

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

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

$4 (h + 2 x)$

Apply the distributive property.

$4 h + 4 (2 x)$

Simplify the expression.

Multiply $2$ by $4$ .

$4 h + 8 x$

Reorder $4 h$ and $8 x$ .

$8 x + 4 h$

Step 5

Find the Difference Quotient f(x)=4x^2+5

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