Using Differentiate from the first principles

Question

y=x^3-2x+1

Answer 1

y(x)=x³-2x+1;

y(x+h)=(x+h)³-2(x+h)+1=x³+3hx²+3h²x-2x-2h+1;

y(x+h)-y(x)=3hx²+3h²x-2h.

The gradient or slope between x and x+h is (3hx²+3h²x-2h)/(x+h-x)=3x²+3hx-2.

As h→0 this expression becomes 3x²-2, which is the differential of x³-2x+1 with respect to x.