- #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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- Thread starter Will
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- #1

Will

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

- #2

enigma

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Do laplace transforms on it.

- #3

Hurkyl

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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

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

- #6

Hurkyl

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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)
```

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

ahrkron

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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.

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