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QuestionSimplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} for f(x)=9x+4f(x)=9x+4, where h≠0h \neq 0.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=9x+4f(x) =9x +4 . We are to find the difference quotient, which is defined as f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}, where h≠0h \neq0.

STEP 2

First, we need to find the value of the function f(x+h)f(x+h). This can be done by replacing xx with (x+h)(x+h) in the function.
f(x+h)=9(x+h)+4f(x+h) =9(x+h) +4

STEP 3

implify the expression f(x+h)f(x+h).
f(x+h)=9x+9h+f(x+h) =9x +9h +

STEP 4

Now, we will find the difference quotient by substituting f(x+h)f(x+h) and f(x)f(x) into the difference quotient formula.
f(x+h)−f(x)h=(9x+9h+4)−(9x+4)h\frac{f(x+h)-f(x)}{h} = \frac{(9x +9h +4) - (9x +4)}{h}

STEP 5

implify the numerator of the difference quotient.
f(x+h)−f(x)h=9hh\frac{f(x+h)-f(x)}{h} = \frac{9h}{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=9\frac{f(x+h)-f(x)}{h} =9The difference quotient of the function f(x)=9x+4f(x) =9x +4 is 99.

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