How do you take this derivative?

1. Oct 17, 2012

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.

2. Oct 17, 2012

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'?

3. Oct 17, 2012

msell2

So everything is the same even though there are vectors?

4. Oct 18, 2012

Ray Vickson

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

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

Last edited: Oct 18, 2012