Question

Please I am lost on how to find the derivative of function using first principal,could someone please work out an example on how to solve such problem?

Answer 1

Example:

f(x)=x²

f(x+dx)
=(x+dx)³
=x³+3x²dx+3x(dx)²+(dx)³

[f(x+dx)-f(x)]/dx
=[x³+3x²dx+3x(dx)²+(dx)³ - x³]/dx
=[3x²dx+3x(dx)²+(dx)³]/dx
=3x²+3x(dx)+(dx)²

Take limit when dx->0
f'(x)
=[f(x+dx)-f(x)]/dx
=3x²

Note: For simplicity, above used dx instead of δx.
δx is meant to be finite, and
dx is generally taken to be infinitely small (after taking limit of δx->0)

Answer 2

correction:
problem is f(x)=x³