What is $f '(x)$ for $f(x)=(x-3)^3$? 
What is $f '(x)$ for $f(x)=(x-3)^3$?

I'm thinking it is $3x^2 - 18x + 27$
but my textbook says it is $3x^2 - 18x - 27$

Using the chain rule we have:
$f'(x)=3(x-3)^2\times \frac{d}{dx}(x-3) = 3(x-3)^2$
And $3(x-3)^2 = 3(x^2-6x+9) = 3x^2-18x+27$
You are right. Your textbook is wrong.
