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- Summary:
- a function which includes step functions, but the step function changes with time

I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me?

##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\

I(R_j) = \begin{array}{cc}

\{ &

\begin{array}{cc}

0 & R_j < 7 \\

1 & R_j \geq 7

\end{array}

\end{array} \\

a_j \text{ is some known constant}

##

##R_j(t) : t \rightarrow Z## (##R_j## is an integer ##\in [1,10]## which decays over time)

##R_j## starts at ##10## and decays over time, and I can approximate its derivative as : ##\frac{dR_j}{dt}= \frac{R_j(t_2) - R_j(t_1)}{t_2-t_1}##

I would like to take the derivative of ##S##

##\frac{\partial S}{\partial t} = \sum_{j=1}^n (\frac{\partial I_j}{\partial R_j} \frac{\partial R_j}{\partial t})a_j##

But, I read that the derivative of a shifted Heaviside function ##H(x-7)## is the dirac delta function ##\delta(x-7)##. But, this dirac delta function, as I am aware, has an infinite value at ##x=7##.

But, the change in ##S## is definitely finite because ##R## decays linearlly with time, so how can ##S## be infinite? How can I get the derivative of ##S##? I want to show how ##S## changes as a function of ##R##

I think the unit step is just by nature not continuous... is there an approximation I can use for this function that would be continuous?

##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\

I(R_j) = \begin{array}{cc}

\{ &

\begin{array}{cc}

0 & R_j < 7 \\

1 & R_j \geq 7

\end{array}

\end{array} \\

a_j \text{ is some known constant}

##

##R_j(t) : t \rightarrow Z## (##R_j## is an integer ##\in [1,10]## which decays over time)

##R_j## starts at ##10## and decays over time, and I can approximate its derivative as : ##\frac{dR_j}{dt}= \frac{R_j(t_2) - R_j(t_1)}{t_2-t_1}##

I would like to take the derivative of ##S##

##\frac{\partial S}{\partial t} = \sum_{j=1}^n (\frac{\partial I_j}{\partial R_j} \frac{\partial R_j}{\partial t})a_j##

But, I read that the derivative of a shifted Heaviside function ##H(x-7)## is the dirac delta function ##\delta(x-7)##. But, this dirac delta function, as I am aware, has an infinite value at ##x=7##.

But, the change in ##S## is definitely finite because ##R## decays linearlly with time, so how can ##S## be infinite? How can I get the derivative of ##S##? I want to show how ##S## changes as a function of ##R##

I think the unit step is just by nature not continuous... is there an approximation I can use for this function that would be continuous?

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