Full Limit Derivatives?

Question

I need to use the full limit definition of the derivative to find the derivative of this function:

f(x) = 2x + 3

Please SHOW WORK!!!

Answer 1

I'm guessing you mean the definition of a derivative? Otherwise, I'm clueless.
The definition of a derivative is
lim f(x+h)-f(x)
h->0 h

So plugging that in, we get:
2(x+h)+3 - (2x+3)
---------------------
h
2x+2h+3-2x-3
___________
h
2h
____
h
=2

Answer 2

f'(x) = Lim [f(x+h)- f(x)] /h
h->0
f(x) = 2x + 3
f(x+h) = [2(x+h)] + 3
f(x+h) - f(x) = [2(x+h)] + 3 - [2x + 3]
f(x+h) - f(x) = [2x + 2h + 3 - 2x -3]
f(x+h) - f(x) = [2h]
Lim [f(x+h) - f(x)] /h = 2
h->0
Thus, f'(x) = 2