Question

Using the limit definition, how do you find the derivative of `y = x^2 + x + 1`?

Answer 1

`lim h ->0 ( [(a+h)^2+(a+h)+1]-[a^2+a+1))/h`
`lim h->0 (a^2+2ah+h^2+a+h+1-a^2-a-1)/h = lim h ->0 (2ah+h^2+h)/h-> lim h ->0 (h(2a+h+1))/h = 2a +1`

Explanation:

Find `f(a+h) = (a+h)^2+(a+h)+1 = a^2+2ah+h^2+a+h+1` and `f(a) =a^2+a+1` then plug it in to the formula `lim h ->0 ( f(a+h)-f(a))/h` and then simplify. Remember to put in 0 for h after you factor out h from the numerator and cancel it to the h in the denominator. The remaining terms are your answer.