Math Snap

STEP 1

Assumptions1. The function is $f(x)=x^{}+8 x+5$
. We are asked to find and simplify the difference quotient of $f$, $\frac{f(x+h)-f(x)}{h}$, where $h \neq0$

STEP 2

First, we need to find $f(x+h)$. We can do this by replacing every $x$ in the function $f(x)$ with $(x+h)$.
$$f(x+h) = (x+h)^{2}+8 (x+h)+5$$

STEP 3

Now, expand the expression $(x+h)^{2}$ and simplify the equation.
$$(x+h)^{2} = x^{2} +2xh + h^{2}$$$$f(x+h) = x^{2} +2xh + h^{2} +8x +8h +5$$

STEP 4

Next, we need to find $f(x+h)-f(x)$. This can be done by subtracting $f(x)$ from $f(x+h)$.
$$f(x+h)-f(x) = [x^{2} +2xh + h^{2} +8x +8h +] - [x^{2} +8x +]$$

STEP 5

implify the above expression by cancelling out common terms.
$$f(x+h)-f(x) =2xh + h^{2} +8h$$

STEP 6

Now, we can find the difference quotient, $\frac{f(x+h)-f(x)}{h}$, by dividing $f(x+h)-f(x)$ by $h$.
$$\frac{f(x+h)-f(x)}{h} = \frac{2xh + h^{2} +8h}{h}$$

STEP 7

implify the above expression by factoring out $h$ from the numerator.
$$\frac{f(x+h)-f(x)}{h} = \frac{h(2x + h +)}{h}$$

Cancel out the $h$ in the numerator and the denominator.
$$\frac{f(x+h)-f(x)}{h} =2x + h +8$$The difference quotient of the function $f(x)=x^{2}+8 x+5$ is $2x + h +8$.