Finding the Derivative

Question

Find the difference quotation f(x) - f(2) / x - 2 when f(x) = 2 + 2x - 4x^2. Simplify the expression fully as if you were going to compute the limit x -> 2. In particular, cancel common factors of x - 2 in the numerator and denominator if possible.

f(x) - f(2)/ x - 2 = -4x^2 + 2x +12/ x-2

Use your result to compute the derivative.

f’(2) = ?

What would f’(2) equal to?

Answer 1

f(x) = 2+2x -4x^2

f'(x) = 2-8x

f'(2) = 2-8(2) = 2-16

f'(2) =-14

That's the derivative using shortcut rules. If you do the difference quotient and limit, and get a different answer, you made a mistake.

[f(x+h)- f(x)]/h = [2+2(x+h) -4(x+h)^2 - 2-2x +4x^2]/h

= [2h -8xh +h^2]/h = 2-8x +h

as h approaches zero 2-8x+h approaches 2+8x

for x=2, 2-8x = 2-8(2)= -14 = f'(2)

or the way you have this set up

[f(x)-f(2)]/(x-2) = [2+2x-4x^2 - (2+2(2)-4(2)^2]/(x-2)

= (2x-4x^2+12)/(x-2) = -2(2x^2+x-6)/(x-2) =-2((2x+3)(x-2)/(x-2) = -2(2x+3) = -2(2(2)+3) = -2(7) =-14