Derivative of unit step function

In summary, the derivative of a unit step function can be obtained by using Laplace transforms or Fourier transforms. However, it is important to note that the derivative of a step function should be treated as a delta function and used as a distribution. Additional requirements, such as integrability and continuity, may also apply.
  • #1
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' ??
 
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  • #2
Do laplace transforms on it.
 
  • #3
Write out the definition of the unit step function and it might be easier to see.
 
  • #4
I think I got it now. I used the property L{f'}(s) = sL{f}(s) - f(0)
Is that correct?
 
  • #5
The Fourier Transform

can also be used. It can be used for many unbounded functions.
 
  • #6
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:
  • #7
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.
 

Related to Derivative of unit step function

1. What is the definition of the derivative of a unit step function?

The derivative of a unit step function is a mathematical concept that represents the rate of change of the unit step function with respect to its input variable. It is defined as the limit of the change in the function's output divided by the change in its input as the change in the input approaches zero.

2. How do you find the derivative of a unit step function?

To find the derivative of a unit step function, you can use the definition of the derivative or apply the derivative rules for step functions. These rules state that the derivative of a step function is zero everywhere except at the step, where it is undefined.

3. What is the value of the derivative of a unit step function at the step?

The value of the derivative of a unit step function at the step is undefined. This is because the step function is discontinuous at the step, and the derivative is not defined for discontinuous functions.

4. Is the derivative of a unit step function a continuous function?

No, the derivative of a unit step function is not a continuous function. This is because the unit step function is discontinuous at the step, and the derivative is not defined at that point. However, the derivative is continuous everywhere else.

5. What is the physical interpretation of the derivative of a unit step function?

The physical interpretation of the derivative of a unit step function is the instantaneous rate of change of a quantity that changes abruptly at a specific point. This can be seen in real-world scenarios such as measuring the speed of an object that suddenly changes direction at a certain time.

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