Question

Using the limit definition, how do you differentiate `f(x)=x^2+3x+1`?

Answer 1

`f'(x)=2x+3`

Explanation:

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

In this case

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

`=lim_(hrarr0)((x+h)^2+3(x+h)+1-x^2-3x-1)/h`

`=lim_(hrarr0)(x^2+2xh+h^2+3x+3h+1-x^2-3x-1)/h`

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

`=lim_(hrarr0)2x+h+3=2x+3`.