How do you take this derivative?

[msell2]

x(t) = (Cv + D(tv + w))e^(λt)
x'(t) = ?
C, D, λ are constants
x(t), v, w are vectors (v is an eigenvector, w is a generalized eigenvector)
t is the variable

This is coming from a larger question, asking to prove that the above equation satisfies x'(t)=Ax(t), where A is a 2x2 matrix and x'(t) and x(t) are vectors. If you know how to answer this too, that'd be great. I've been trying to figure this out for some time now. Any help would be appreciated.

 

[HallsofIvy]

x is the product of two functions, f(t)= (Cv + D(tv + w)) and g(t)= e^(λt). Do you know how to differentiate each of those? Do you know the product rule: (fg)'= f'g+ fg'?

 

[msell2]

So everything is the same even though there are vectors?

 

[Ray Vickson]

So everything is the same even though there are vectors?

Sure: just write it out component-by-component:
$$x_1(t) = (Cv_1 + D(v_1 t + w_1))e^{\lambda t}\\<br /> x_2(t) = (Cv_2 + D(v_2 t + w_2))e^{\lambda t}\\<br /> \vdots \\<br /> x_n(t) = (Cv_n + D(v_n t + w_n))e^{\lambda t}$$

Just for practice (and to help clarify some of the issues), try to see what would happen if C and D are constant n×n matrices instead of scalar constants.

RGV