$\left(\mathbf{A}·\mathbf{B}\right)\prime$

$\=\left(a\left(x\right)u\left(x\right)\+b\left(x\right)v\left(x\right)\right)\prime$

$\=\left(a\left(x\right)u\left(x\right)\right)\prime \+\left(b\left(x\right)v\left(x\right)\right)\prime$

$\=\left(a u\prime \+u a\prime \right)\+\left(b v\prime \+v b\prime \right)$

$\=\left(a u\prime \+b v\prime \right)\+\left(u a\prime \+v b\prime \right)$

$\=\mathbf{A}·\mathbf{B}\prime \+\mathbf{B}·\mathbf{A}\prime$

Loading Student:-MultivariateCalculus

Define the vectors A and B 

$⟨a\left(x\right)\,b\left(x\right)⟩$$\stackrel{\text{assign to a name}}{\to }$$A$

$⟨u\left(x\right)\,v\left(x\right)⟩$$\stackrel{\text{assign to a name}}{\to }$$B$

Show the difference between $\left(\mathbf{A}·\mathbf{B}\right)\prime$ and $\mathbf{A}·\mathbf{B}\prime \+\mathbf{B}·\mathbf{A}\prime$ is zero

$\left(\mathbf{A}·\mathbf{B}\right)\prime -\left(\mathbf{A}·\mathbf{B}\prime \+\mathbf{B}·\mathbf{A}\prime \right)$ = $0$

Alternate notation and calculation

$\frac{\ⅆ}{\ⅆ x} \left(\mathbf{A}·\mathbf{B}\right)-\left(\mathbf{A}·\frac{\ⅆ}{\ⅆ x} \mathbf{B}\+\mathbf{B}·\frac{\ⅆ}{\ⅆ x} \mathbf{A}\right)$ = $0$