Supposef(x) =x2+ 2x. Findf(x+h)−f(x)h.

Question

Suppose f(x) = x2+ 2x. Find f(x+h)−f(x)/h.

Answer 1

f(x+h) = (x+h)^2 + 2(x+h) = x^2 +2xh + h^2 +2x + 2h

subtract x^2 + 2x

leaving: 2xh + h^2 +2h

divide by h to get 2x + h + 2

this is all preparation to take the limit as h goes to zero, which leaves 2x +2, which is the derivative of f(x).

f'(x) is the derivative or slope of the quadratic curve at the point (x, f(x)) Later you learn a short cut rule to find the derivative. For x^2 +2x, bring the exponent down as a coefficient, and reduce the exponent by 1, to get 2x. derivative of 2x is just 2. So, 2x+2 is the derivative of x^2 +2x. h is a change in x, then take the limit as h goes to zero

if you work out the solution with [f(x+h) - f(x)]/h where h goes to zero, and get something different from the short cut rule, you made a mistake somewhere