Question

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

Answer 1

The derivative of a function in a point `x` is the:

`lim_(h rarr 0) (f(x+h)-f(x))/h`

Explanation:

Thus we evaluate:

`lim_(h rarr 0) (4(x+h)^2-4x^2)/h=lim_(h rarr 0) (4(x^2+2xh+h^2)-4x^2)/h=lim_(h rarr 0) (4x^2+8xh+4h^2-4x^2)/h=lim_(h rarr 0) (8xh+4h^2)/h=lim_(h rarr 0) (h(8x+4h))/h=lim_(h rarr 0) (8x+4h)=8x`

So `(4x^2)'=8x`