Question

Using the limit definition, how do you differentiate `f(x)=3-x^2`?

Answer 1

The limit definition of the derivative:

`lim_(hrarr0)(f(x+h)-f(x))/h`

We know that `f(x)=3-x^2` and
`f(x+h)=3-(x+h)^2`

`=lim_(hrarr0)(3-(x+h)^2-(3-x^2))/h`

`=lim_(hrarr0)(3-(x^2+2hx+h^2)-3+x^2)/h`

`=lim_(hrarr0)(3-x^2-2hx-h^2-3+x^2)/h`

`=lim_(hrarr0)(-2hx-h^2)/h`

`=lim_(hrarr0)-2x-h`

Now we can plug in `0` for `h`.

`=-2x`

Thus, `f'(x)=-2x`.