`f(x) = 6/x` Find the derivative of the function by the limit process.

Expert Answers
sciencesolve eNotes educator| Certified Educator

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).`