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QuestionSimplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} for f(x)=x2+8x+5f(x)=x^{2}+8x+5, where h≠0h \neq 0.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x+8x+5f(x)=x^{}+8 x+5 . We are asked to find and simplify the difference quotient of ff, f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}, where h≠0h \neq0

STEP 2

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

STEP 3

Now, expand the expression (x+h)2(x+h)^{2} and simplify the equation.
(x+h)2=x2+2xh+h2(x+h)^{2} = x^{2} +2xh + h^{2}f(x+h)=x2+2xh+h2+8x+8h+5f(x+h) = x^{2} +2xh + h^{2} +8x +8h +5

STEP 4

Next, we need to find f(x+h)−f(x)f(x+h)-f(x). This can be done by subtracting f(x)f(x) from f(x+h)f(x+h).
f(x+h)−f(x)=[x2+2xh+h2+8x+8h+]−[x2+8x+]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+h2+8hf(x+h)-f(x) =2xh + h^{2} +8h

STEP 6

Now, we can find the difference quotient, f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}, by dividing f(x+h)−f(x)f(x+h)-f(x) by hh.
f(x+h)−f(x)h=2xh+h2+8hh\frac{f(x+h)-f(x)}{h} = \frac{2xh + h^{2} +8h}{h}

STEP 7

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

STEP 8

Cancel out the hh in the numerator and the denominator.
f(x+h)−f(x)h=2x+h+8\frac{f(x+h)-f(x)}{h} =2x + h +8The difference quotient of the function f(x)=x2+8x+5f(x)=x^{2}+8 x+5 is 2x+h+82x + h +8.

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