# Derivative of unit step function

Will
[SOLVED] Derivative of unit step function

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

enigma
Staff Emeritus
Gold Member
Do laplace transforms on it.

Hurkyl
Staff Emeritus
Gold Member
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?

The Fourier Transform

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

Hurkyl
Staff Emeritus
Gold Member
You could just differentiate it directly.

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

is equivalent to:

Code:
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:

Code:
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

Last edited:
ahrkron
Staff Emeritus
Gold Member
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.