Help me with this Derivative Question (I have the answer but don't know how to solve it)?

Question

The Question is:
What is the value of lim((x + "delta" x)^3 - x^3)/ "delta" x
if x=3
The answer is 3x^2 and thus 27.
How do you arrive at this answer? Can someone explain to me how to solve these kinds of questions?

Answer 1

For a question like this you need to expand out the (x+Δx)³ term:
(x+Δx)³ - x³ = x³ + 3x²Δx + 3x(Δx)² + (Δx)³ - x³ = 3x²Δx + 3x(Δx)² + (Δx)³

The goal is to get a factor of Δx in the numerator, with which you can cancel the Δx in the denominator; then you will be able to let Δx -> 0. As you can see, this factoring can be done in the current problem:

3x²Δx + 3x(Δx)² + (Δx)³ = Δx(3x² + 3x Δx + (Δx)² )

So

[(x+Δx)³ - x³]/Δx = Δx(3x² + 3x Δx + (Δx)² )/Δx = 3x² + 3x Δx + (Δx)²

Then as Δx -> 0, this last expression goes to 3x²

Answer 2

4^2