Question

How do you find f'(x) using the limit definition given `f (x) = -(2/3x)`?

Answer 1

`- 2/3`

Explanation:

the classic definition is `f'(x) = (f(x+h) - f(x) )/ h |_{h \to 0}`

here that becomes

`f'(x) =( - (2/3 (x + h)) -( - 2/3 x) )/ h |_{h \to 0}`

`= (- (2/3 h) )/ h |_{h \to 0}`

`= - 2/3`

which is what you would expect :-)

crucially because we are in the limit `h \to 0` but `h \ne 0` we can do that division and cancel the h's.