- #1

Will

## Main Question or Discussion Point

**[SOLVED] Derivative of unit step function**

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

- Thread starter Will
- Start date

- #1

Will

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

- #2

enigma

Staff Emeritus

Science Advisor

Gold Member

- 1,747

- 10

Do laplace transforms on it.

- #3

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

- 14,916

- 19

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

- #4

Will

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

Is that correct?

Is that correct?

- #5

- 398

- 0

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

- #6

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

- 14,916

- 19

You could just differentiate it directly.

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

is equivalent to:

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

Differentiating on each piece gives:

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

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

is equivalent to:

Code:

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

Differentiating on each piece gives:

Code:

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

IOW:

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

Last edited:

- #7

ahrkron

Staff Emeritus

Gold Member

- 736

- 1

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.

- Last Post

- Replies
- 7

- Views
- 7K

- Last Post

- Replies
- 1

- Views
- 3K

- Replies
- 1

- Views
- 984

- Replies
- 2

- Views
- 689

- Last Post

- Replies
- 3

- Views
- 4K

- Last Post

- Replies
- 5

- Views
- 2K

- Last Post

- Replies
- 2

- Views
- 14K

- Last Post

- Replies
- 5

- Views
- 960

- Last Post

- Replies
- 3

- Views
- 3K

- Replies
- 1

- Views
- 762