the difference quotient of 2x^2+3x-1

Question

i dont understand how to get the answer for this problem

Answer 1

Let f(x)=2x²+3x-1, so f(x+h)=2(x+h)²+3(x+h)-1=2x²+4hx+2h²+3x+3h-1.

The difference quotient is (f(x+h)-f(x))/h=

(2x²+4hx+2h²+3x+3h-1-(2x²+3x-1))/h=

(2x²+4hx+2h²+3x+3h-1-2x²-3x+1)/h= (matching colour terms cancel out)

(4hx+2h²+3h)/h=4x+2h+3.

When h becomes 0 the difference quotient becomes 4x+3, which is the derivative or differential of f(x) with respect to x.