Am I finding the derivative right

The question is find the derivative of f(x)= (x-2)(x-3)^3

I used the product rule:
so f '(x) = (x-2) d/dx(x-3)^2 + (x-3)^2 d/dx(x-2)
= (x-2)(d/dx x^2 - 6x + 9) +(x-3)^2 (1)
= (x-2)(2x -6) + x^2 - 6x +9
= 2x^2 -6x -4x +12 +x^2 - 6x +9
= 3x^2 - 16x+ 21

thank you :-) for taking the time to read this,

-a frustrated calculus student

Let f(x) = (x-2) and g(x) = (x-3)^2

(f(x)g(x))' = (1) * (x-3)^2 + (x-2) * [2 * (x-3)]

= (x-3)^2 + [(x-2) * [2x-6]

= (x-3)^2 + 2x^2 - 6x -4x + 12

= x^2 - 6x + 9 + 2x^2 - 6x -4x + 12

= 3x^2 -16x + 21