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

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=4xf(x)=4x . We are asked to find and simplify 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) and f(x)f(x).For f(x)f(x), we simply substitute xx into the function.
f(x)=4xf(x) =4xFor f(x+h)f(x+h), we substitute x+hx+h into the function.
f(x+h)=4(x+h)f(x+h) =4(x+h)

STEP 3

Now, we substitute f(x+h)f(x+h) and f(x)f(x) into the difference quotient formula.
f(x+h)−f(x)h=(x+h)−xh\frac{f(x+h)-f(x)}{h} = \frac{(x+h) -x}{h}

STEP 4

Next, we simplify the numerator by distributing 44 in 4(x+h)4(x+h) and subtracting 4x4x.
f(x+h)−f(x)h=4x+4h−4xh\frac{f(x+h)-f(x)}{h} = \frac{4x +4h -4x}{h}

STEP 5

implify the numerator further by cancelling out 4x4x.
f(x+h)−f(x)h=4hh\frac{f(x+h)-f(x)}{h} = \frac{4h}{h}

STEP 6

Finally, we simplify the difference quotient by cancelling out hh.
f(x+h)−f(x)h=4\frac{f(x+h)-f(x)}{h} =4The difference quotient for the function f(x)=4xf(x) =4x is 44.

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