Question

Solve the difference quotient for the given function

f(x) = x^3, (f(a+h)-f(a))/h

i forgot how to do this problem

Answer 1

Divide [(a+h)^3 - a^3] by h.

Begin by multiplying out the (a+h)^3 term.
(a+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3.

After two more steps,
(f(a+h)-f(a))/h = 3a^2 + 3ah + h^2

As h-> 0, it approaches 3 a^2, the derivative at x=a.

Answer 2

You will need
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

so (f(x+h) - f(x) )/h
= (x^3 + 3x^2h + 3xh^2 + h^3 - x^3)/h
= 3x^2 +3xh + h^2