Question

How do you find the derivative of `F(x)=x^3−7x+5` using the limit definition?

Answer 1

`f'(x)=3x^2-7`

Explanation:

differentiating using the `color(blue)"limit definition"`

`f'(x)=lim_(hto0)(f(x+h)-f(x))/h`

The aim here is to obtain a factor h on the numerator which will cancel the h on the denominator.

`f'(x)=lim_(hto0)((x+h)^3-7(x+h)+5-(x^3-7x+5))/h`

`lim_(hto0)(cancel(x^3)+3x^2h+3xh^2+h^3cancel(-7x)-7hcancel(+5)cancel(-x^3)cancel(+7x)cancel(-5))/h`

`=lim_(hto0)(cancel(h)(3x^2+3xh+h^2-7))/cancel(h)`

The terms with an h on the numerator `=0" as "hto0`

`rArrf'(x)=3x^2-7`