Question

How do you find f'(x) using the definition of a derivative for `f(x)=(4/x^2)`?

Answer 1

See the explanation.

Explanation:

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

`f'(x)=lim_(h->0) (4/(x+h)^2-4/x^2)/h = lim_(h->0) ((4x^2-4(x+h)^2)/(x^2(x+h)^2))/h`

`f'(x)= lim_(h->0) (4x^2-4x^2-8xh-4h^2)/(hx^2(x+h)^2)`

`f'(x)= lim_(h->0) (-8xh-4h^2)/(hx^2(x+h)^2) = lim_(h->0) (-8x-4h)/(x^2(x+h)^2) = (-8x)/(x^2x^2)`

`f'(x)= -8/x^3`