Question

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

Answer 1

`f'(x) = 3x^2`

Explanation:

By definition of the derivative `f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h`
So with `f(x) = x^3` we have;

`f'(x) = lim_(h rarr 0) ( (x+h)^3 - x^3 ) / h`
`:. f'(x) = lim_(h rarr 0) ( x^3+3x^2h+3xh^2+h^3 - x^3 ) / h`
`:. f'(x) = lim_(h rarr 0) ( 3x^2h+3xh^2+h^3 ) / h`
`:. f'(x) = lim_(h rarr 0) ( 3x^2+3xh+h^2 )`
`:. f'(x) = 3x^2`