How do you derive using first principles?

Question

Simple question i guess, can you just go through the steps involved using first principles?

Answer 1

f `(x) = lim h->0 [ f(x + h) - f(x) ] / h-------h ≠ 0
Example
Differentiate 3x² from 1st principles.
f `(x) = lim h->[ 3.(x + h)² - 3x² ] / h
f `(x) = lim h->0 [3.(x² + 2hx + h²) - 3x² ] / h
f `(x) = lim h->0 [ 6hx + h² ] / h
f `(x) = lim h->0 (6x + h)
f `(x) = 6x

Hope this helps.

Answer 2

I will try and show you by an example for the curve y = x²

First of all we need the idea of a small change, letter δ.

We look at the point on a curve (x, y) and make a small change in x, that is x + δx.

Next we look at the effect of this on y and there will be a small change in y, called δy.

now both these points lie on y = x² so

For (x + δx , y + δy) ..........y + δy = (x + δx)²

and for (x , y)......................y ......... = x²

subtracting the two.................... δy = (x + δx)² - x²

so δy = x² + 2xδx + (δx)² - x²

.... δy = 2xδx + (δx)²

... δy / δx = 2x + δx ............ dividing both sides by δx

Lastly we let δx ----> 0

dy/dx = 2x

Answer 3

Simple answer, nothing to derive, nothing to show.