Question

How do you find the derivative of `f(x)=4x^2` using the limit definition?

Answer 1

`f'(x)= 8x`

Explanation:

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

So for `f(x)=4x^2` we have

`f'(x)= lim_(h->0)({4(x+h)^2 - 4x^2})/h`
`:. f'(x)= lim_(h->0)({4(x^2+2hx+h^2) - 4x^2})/h`
`:. f'(x)= lim_(h->0)({4x^2+8hx+4h^2 - 4x^2})/h`
`:. f'(x)= lim_(h->0)(8hx+4h^2 )/h`
`:. f'(x)= lim_(h->0)(8x+4h )`
`:. f'(x)= 8x`