Calculus of a Single Variable, Chapter 2, Review, Section Review, Problem 3

You need to find derivative using limit definition, such that:
f'(x)= lim_(Delta x -> 0) (f(x + Delta x) - f(x))/(Delta x)
f'(x) = lim_(Delta x -> 0) ((x + Delta x)^2 - 4(x+Delta x) + 5 - x^2 + 4x - 5)/(Delta x)
f'(x) = lim_(Delta x -> 0) (x^2 + 2x*Delta x + Delta^2 x - 4x - 4 Delta x + 5 - x^2 + 4x - 5)/(Delta x)
Reducing like terms yields:
f'(x) = lim_(Delta x -> 0) (2x*Delta x + Delta^2 x - 4 Delta x)/(Delta x)
Simplify by Delta x :
f'(x) = lim_(Delta x -> 0) 5(2x + Delta x - 4)
Replacing 0 for Delta x yields:
f'(x) = 2x - 4
Hence, evaluating the limit of function using limit definition, yields f'(x) = 2x - 4.