Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=3x^2`?

Answer 1

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

`f'(x) = lim_(h-> 0) (3(x + h)^2 - 3x^2)/h`

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

`f'(x) = lim_(h-> 0) (3x^2 + 6xh + 3h^2 - 3x^2)/h`

`f'(x) = lim_(h-> 0) (6xh + 3h^2)/h`

`f'(x) = lim_(h-> 0) (h(6x+ 3h))/h`

`f'(x)= lim_(h-> 0) 6x + 3h`

`f'(x) = 6x + 3(0)`

`f'(x) = 6x`

Verification using the power rule yields the same result.

Hopefully this helps!