Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=4/(x-3)`?

Answer 1

The limit definition is the formula `f'(x) = lim_(h -> 0) (f(x + h) - f(x))/h`.

`f'(x) = lim_(h->0)(4/(x + h - 3) - 4/(x - 3))/h`

This involves lots of algebra--brace yourself!

`f'(x) = lim_(h->0)((4(x - 3))/((x + h -3)(x - 3)) - (4(x + h - 3))/((x - 3)(x + h - 3)))/h`

`f'(x) = lim_(h->0) ((4x - 12 - 4x - 4h + 12)/((x + h -3)(x- 3)))/h`

`f'(x) = lim_(h->0) ((-4h)/((x + h - 3)(x - 3)))/h`

`f'(x) = lim_(h->0)(-4h)/((x + h - 3)(x - 3)(h))`

`f'(x) = lim_(h->0)(-4)/((x + h - 3)(x - 3))`

We can now substitute `h =0` into the expression.

`f'(x) = -4/((x + 0 - 3)(x- 3))`

`f'(x) = -4/((x - 3)(x - 3))`

`f'(x) = -4/(x - 3)^2`

Hopefully this helps!