Question

Using the limit definition, how do you differentiate `f(x) =9-x^2`?

Answer 1

The derivative is defined as the limit of the incremental ratio, so:

`f'(x) = lim_(Delta x ->0) (f(x+Delta x) - f(x))/(Delta x)`

Explanation:

`(Delta f(x))/(Delta x) = (9 - (x+Delta x)^2 - 9 + x^2)/(Delta x)`

`(Delta f(x))/(Delta x) = (cancel(9) cancel (- x^2) -2xDelta x-(Delta x)^2cancel(- 9) + cancel(x^2))/(Delta x) =-2x- Delta x`

Carrying to the limit:

`lim_(Delta x ->0) (Delta f(x))/(Delta x) = -2x`