The Product Rule AB & BC

The derivative of a product is $\frac{d}{dx}[uv] = u'v + uv'$: two terms, each differentiating exactly one factor and copying the other. There is no simpler rule, and $u = v = x$ shows why, since $(x \cdot x)' = 2x$ while $u'v' = 1$. Writing $u$, $v$, $u'$ and $v'$ down before assembling keeps both terms alive, and when both factors are powers of $x$ expanding first is faster and doubles as a check.

The dominant error is differentiating the factors separately and multiplying the results, or applying the rule and then dropping one of its two terms. Alongside that, the constant multiple case gets tangled: a numerical factor treated as a full second function, or a coefficient lost while the two terms are being assembled, particularly when a minus sign arrives in the middle of the expression from a cosine.