[SOLVED] Derivative of unit step function

[Will]

How does one do this, for example x= e^(-3t)u(t-4); how do you get x' ??

[enigma]

Do laplace transforms on it.

[Hurkyl]

Write out the definition of the unit step function and it might be easier to see.

[Will]

I think I got it now. I used the property L{f'}(s) = sL{f}(s) - f(0)
Is that correct?

[Tyger]

can also be used. It can be used for many unbounded functions.

[Hurkyl]

You could just differentiate it directly.

x(t) = e^(-3t)u(t-4)

is equivalent to:


x(t) = e^(-3t) (for t > 4)
0 (for t < 4)


with x(4) depending on the precise definition of u.

Differentiating on each piece gives:


x'(t) = (-3) e^(-3t) (for t > 4)
0 (for t < 4)


And x'(4) does not exist because x(t) is discontinuous at t = 4

IOW:

x'(t) = (-3) e^(-3t) u(t - 4) for t [x=] 4

[ahrkron]

Sometimes you can safely assume the derivative of a step to be a delta function (for instance, when you integrate a delta, you get a step).

They need to be used as distributions, and there may be some requirements on the functions you use along with them (integrability, continuity,...).

I'm sorry I don't remember much about it.