How would I differentiate something with the product rule?

Okay so, first of all the product rule in itself is fairly simple. Take the product of a function "u" and a function "v" to be a compound function. You can write this as "uv". The standard result is "u'v + v'u", where u' and v' are basically just the differentiated functions u and v. So now lets take the example of x²sin(x) where we know both how to differentiate x^2 and we know how to differentiate sin(x). If we say x² is u and sin(x) is v then we know u' is 2x, and v' is cos(x). Now we have u, u', v and v' we can just substitute them into the standard pattern giving us 2xsin(x) + x²*cos(x). Try the same thing with x³ * cos(x) now.