Calculus of a Single Variable, Chapter 2, Review, Section Review, Problem 4

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) (6/(x+Delta x) - 6/x)/(Delta x)
f'(x) = lim_(Delta x -> 0) (6x - 6x - 6Delta x)/(x*Delta x*(x+Delta x))
Reducing like terms yields:
f'(x) = lim_(Delta x -> 0) (-6Delta x)/(x*Delta x*(x+Delta x))
Simplify by Delta x :
f'(x) = lim_(Delta x -> 0) (-6)/(x*(x+Delta x))
Replacing 0 for Delta x yields:
f'(x) = -6/(x^2)
Hence, evaluating the limit of function using limit definition, yields f'(x) =-6/(x^2).