How/when should I use the product rule for differentiation?

The product rule can be used to differentiate a function that is formed of the product of two other functions;

e.g f(x)=x²e^x

the product rule is as follows; if f(x) is split up into u.v (in this case u would be x² and v would be e^x), the derivative of th whole function is (u.dv/dx) + (v.du/dx)

so in this case u=x², following standard differentiation du/dx= 2x

v=e^x, dv/dx=e^x

u.dv/dx=x²e^x

v.du/dx=2xe^x

so the whole function differentiated = e^x(x²+2x)