Let f be a function that satisfies f(x+𝜀)=f(x) − 9x𝜀 + 9𝜀2 , for any real numbers x and 𝜀 , then f'(x)=

Answer 1

The definition of f'(x) is the limit of (f(x+ε)-f(x))/ε as ε → 0. So we want to write the given expression in that form.

f(x+ε) = f(x) - 9xε +9xε² ⇒ (f(x+ε)-f(x))/ε = -9x + 9xε. Taking the limit ε → 0 gives f'(x) = -9x.