Suppose that f (x+h) - f (x) = - 6hx2-7hx-6h2x-7h2+2h3.

Find f′ (x).

f′ (x) =

By definition,

f' (x) = Lim h->0 (f (x+h) - f (x)) / h

We are already given

f (x+h) - f (x) = - 6hx2-7hx-6h2x-7h2+2h3=h (-6x^2-7x-6hx-7h+2h^2)

divide by h

(f (x+h) - f (x)) / h = h (-6x^2-7x-6hx-7h+2h^2) / h = (-6x^2-7x-6hx-7h+2h^2)

Finally, take lim h->0

f' (x) = Lim h->0 (f (x+h) - f (x)) / h = (-6x^2-7x-0-0+0) = - 6x^2-7x

=>

f' (x) = - 6x^2-7x