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QuestionFind the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} for f(x)=−6x+5f(x)=-6x+5, with h≠0h \neq 0. Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=−6x+5f(x)=-6x+5 . We are asked to find the difference quotient 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). This is done by replacing every xx in the function f(x)f(x) with (x+h)(x+h).
f(x+h)=−6(x+h)+5f(x+h) = -6(x+h) +5

STEP 3

implify the expression for f(x+h)f(x+h).
f(x+h)=−6x−6h+5f(x+h) = -6x -6h +5

STEP 4

Now, we can form the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} by substituting f(x+h)f(x+h) and f(x)f(x) into the formula.
f(x+h)−f(x)h=−6x−6h+−(−6x+)h\frac{f(x+h)-f(x)}{h} = \frac{-6x -6h + - (-6x +)}{h}

STEP 5

implify the numerator of the difference quotient.
f(x+h)−f(x)h=−hh\frac{f(x+h)-f(x)}{h} = \frac{-h}{h}

STEP 6

Finally, simplify the difference quotient by cancelling out the hh in the numerator and the denominator.
f(x+h)−f(x)h=−6\frac{f(x+h)-f(x)}{h} = -6The difference quotient of the function f(x)=−6x+5f(x)=-6x+5 is −6-6.

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