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

$\=\frac{d}{\mathrm{dt}}\(\left[\begin{array}{c}{a}_1\(t\)\\ a_2\(t\)\end{array}\right]·\left[\begin{array}{c}{b}_1\(t\)\\ b_2\(t\)\end{array}\right]\)$

$\=\frac{d}{\mathrm{dt}}\left(a_1 b_1\+a_2 b_2\right)$

$\=a_1 b_1^{\mathbf{\/}}\+a_1^{\mathbf{\/}} b_1\+a_2 b_2^{\mathbf{\/}}\+a_2^{\mathbf{\/}} b_2$

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

$\=\left[\begin{array}{c}{a}_1^{\mathbf{\/}}\\ a_2^{\mathbf{\/}}\end{array}\right]·\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]\+\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]·\left[\begin{array}{c}{b}_1^{\mathbf{\/}}\\ b_2^{\mathbf{\/}}\end{array}\right]$

$\=a_1 b_1^{\mathbf{\/}}\+a_1^{\mathbf{\/}} b_1\+a_2 b_2^{\mathbf{\/}}\+a_2^{\mathbf{\/}} b_2$

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

$\=\frac{d}{\mathrm{dt}}\left(\left[\begin{array}{c}{a}_1\(t\)\\ a_2\(t\)\end{array}\right]·\left[\begin{array}{c}{a}_1\(t\)\\ a_2\(t\)\end{array}\right]\right)$

$\=\frac{d}{\mathrm{dt}}\left(a_1^2\left(t\right)\+a_2^2\left(t\right)\right)$

$\=2 a_1{a}_1^{\mathbf{\/}}\+2 a_2{a}_2^{\mathbf{\/}}$

$\=2\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]·\left[\begin{array}{c}{a}_1^{\mathbf{\/}}\\ a_2^{\mathbf{\/}}\end{array}\right]$

$\=2\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]·\(\frac{d}{\mathrm{dt}}\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\)$

$\=2 \mathbf{A}·\left(\frac{d}{\mathrm{dt}}\mathbf{A}\right)$

$\=2 \mathbf{A}·\mathbf{A}\prime$

Loading Student:-MultivariateCalculus

$\mathbf{A}\=⟨a_1\left(t\right)\,a_2\left(t\right)⟩$$\stackrel{\text{assign}}{\to }$

$\mathbf{B}\=⟨b_1\left(t\right)\,b_2\left(t\right)⟩$$\stackrel{\text{assign}}{\to }$

Implement the calculation on the left-hand side of the given identity

$\frac{\ⅆ}{\ⅆ t} \left(\mathbf{A}·\mathbf{B}\right)$

$\left(\frac{\ⅆ}{\ⅆt}{a}_1\left(t\right)\right)b_1\left(t\right)\+a_1\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{b}_1\left(t\right)\right)\+\left(\frac{\ⅆ}{\ⅆt}{a}_2\left(t\right)\right)b_2\left(t\right)\+a_2\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{b}_2\left(t\right)\right)$

Implement the calculation on the right-hand side of the given identity

$\left(\frac{\ⅆ}{\ⅆ t} \mathbf{A}\right)·\mathbf{B}\+\mathbf{A}·\left(\frac{\ⅆ}{\ⅆ t} \mathbf{B}\right)$

$\left(\frac{\ⅆ}{\ⅆt}{a}_1\left(t\right)\right)b_1\left(t\right)\+a_1\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{b}_1\left(t\right)\right)\+\left(\frac{\ⅆ}{\ⅆt}{a}_2\left(t\right)\right)b_2\left(t\right)\+a_2\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{b}_2\left(t\right)\right)$

Implement the calculation on the left-hand side of the given identity

$\frac{\ⅆ}{\ⅆ t} \left(\mathbf{A}·\mathbf{A}\right)$

$2a_1\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{a}_1\left(t\right)\right)\+2a_2\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{a}_2\left(t\right)\right)$

Implement the calculation on the right-hand side of the given identity

$2 \mathbf{A}·\left(\frac{\ⅆ}{\ⅆ t} \mathbf{A}\right)$

$2a_1\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{a}_1\left(t\right)\right)\+2a_2\left(t\right)\left(\frac{\ⅆ}{\ⅆt}{a}_2\left(t\right)\right)$