Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=3x^2+3x+3`?

Answer 1

`f^'(x)=lim_(h->0)(f(x+h)-f(x))/h=lim_(h->0)(3*((x+h)^2+(x+h)+1-(x^2+x+1)))/h=`
`=lim_(h->0)(3*(x^2+2xh+h^2+x+h+1-x^2-x-1))/h`
`=lim_(h->0)(3*(2xh+h^2+h))/h`
`=lim_(h->0)3*(2x+h+1)=3(2x+1)=6x+3`