Find the Difference Quotient f(x)=x^2+5x+3

$f (x) = x^{2} + 5 x + 3$

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.

$f (x + h) = {(x + h)}^{2} + 5 (x + h) + 3$

Simplify the result.

Simplify each term.

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

$f (x + h) = (x + h) (x + h) + 5 (x + h) + 3$

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

Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) + 5 (x + h) + 3$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) + 5 (x + h) + 3$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 5 (x + h) + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h + 5 (x + h) + 3$

Multiply $h$ by $h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} + 5 (x + h) + 3$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} + 5 (x + h) + 3$

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 (x + h) + 3$

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} + 5 x + 5 h + 3$

The final answer is $x^{2} + 2 h x + h^{2} + 5 x + 5 h + 3$ .

$x^{2} + 2 h x + h^{2} + 5 x + 5 h + 3$

Reorder.

Move $5 x$ .

$x^{2} + 2 h x + h^{2} + 5 h + 5 x + 3$

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} + 5 h + 5 x + 3$

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

$h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3$

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3$

$f (x) = x^{2} + 5 x + 3$

$f (x + h) = h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3$

$f (x) = x^{2} + 5 x + 3$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3 - (x^{2} + 5 x + 3)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3 - x^{2} - (5 x) - 1 \cdot 3}{h}$

Simplify.

Multiply $5$ by $- 1$ .

$\frac{h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3 - x^{2} - 5 x - 1 \cdot 3}{h}$

Multiply $- 1$ by $3$ .

$\frac{h^{2} + 2 h x + x^{2} + 5 h + 5 x + 3 - x^{2} - 5 x - 3}{h}$

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

$\frac{h^{2} + 2 h x + 5 h + 5 x + 3 + 0 - 5 x - 3}{h}$

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

$\frac{h^{2} + 2 h x + 5 h + 5 x + 3 - 5 x - 3}{h}$

Subtract $5 x$ from $5 x$ .

$\frac{h^{2} + 2 h x + 5 h + 0 + 3 - 3}{h}$

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

$\frac{h^{2} + 2 h x + 5 h + 3 - 3}{h}$

Subtract $3$ from $3$ .

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

Add $h^{2} + 2 h x + 5 h$ and $0$ .

$\frac{h^{2} + 2 h x + 5 h}{h}$

Factor $h$ out of $h^{2} + 2 h x + 5 h$ .

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

$\frac{h \cdot h + 2 h x + 5 h}{h}$

Factor $h$ out of $2 h x$ .

$\frac{h (h) + h (2 x) + 5 h}{h}$

Factor $h$ out of $5 h$ .

$\frac{h (h) + h (2 x) + h \cdot 5}{h}$

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

$\frac{h (h + 2 x) + h \cdot 5}{h}$

Factor $h$ out of $h (h + 2 x) + h \cdot 5$ .

$\frac{h (h + 2 x + 5)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (h + 2 x + 5)}{h}$

Divide $h + 2 x + 5$ by $1$ .

$h + 2 x + 5$

Reorder $h$ and $2 x$ .

$2 x + h + 5$

Step 5

Find the Difference Quotient f(x)=x^2+5x+3

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