Question

f(x) = x2 + 2x - 1

Compute the derivative (that is, the derivative function) by evaluating the limit. Show your work. (Notice that for the function k(x),
you’ll have to compute the limit differently depending on whether x < 2 or x > 2, and
you’ll end up with a piecewise function for the derivative. Since the graph of k(x) is
made of straight lines, you can also find its derivative by looking at the actual slopes.)

Answer 1

Typical solution for this kind of problem

f(x) = x^2 + 2x + 1
f(x+h) = (x+h)^2 + 2(x+h) + 1
= x^2 + 2hx + h^2 + 2x + 2h + 1

d(f(x))/dx = lim [f(x+h) - f(x)]/h , as h ---> 0
= lim [x^2 + 2hx + h^2 + 2x + 2h + 1 - (x^2 + 2x + 1)]/(x+h - x) , h ---> 0
= lim (x^2 + 2hx + h^2 + 2x + 2h + 1 - x^2 - 2x - 1)/h , as h ---> 0
= lim (2hx + h^2 + 2h)/h as h --->0
= lim 2x + h + 2 , as h --->0
= 2x + 2

no idea where k(x) comes into play