Using the first principle,

Question

Differentiate

(1) x^2+x

(2) 2x^2-3x

Answer 1

Using first principles:

1. Let y=x^2+x. Now increase x by a small value dx which changes y by a small value dy. y+dy=(x+dx)^2+x+dx=x^2+2xdx+dx^2+x+dx. Take y=x^2+x away from this: dy=2xdx+dx^2+dx. Divide through by dx: dy/dx=2x+dx+1. But dx is very small compared to other terms, so can be ignored. That gives us dy/dx=2x+1.
2. Similarly, dy=2(x^2+2xdx+dx^2)-3x-3dx - (2x^2-3x)=4xdx+2dx^2-3dx. Divide by dx: dy/dx=4x-3.

dy/dx is the differential or gradient.