Question

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

f′(x)=

Answer 1

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