How does one verify the derivative of $ f(x) = \sum^{K_2}_{k_2=1} c_{k_2} \exp\left(- \big(a^{(2)} - t^{(2)}_{k_2}\big)^2\right)$

Question

I wanted to verify the derivative of the following equation:

$$ f(x) = \sum^{K_2}_{k_2=1} c_{k_2} \exp\left(- \big(a^{(2)} - t^{(2)}_{k_2}\big)^2\right)$$

for that I wanted to write a symbolic equation/representation of it so that I could manipulate it but I wasn't sure how to do that in mathematica. Ideally I'd like to get the following functionality:

$$ \frac{df(x)}{dt^{(2)}_{k_2}} = c_{k_2} \left(- \exp\big(z^{(2)}_{k_2}\big)\cdot2\big(a^{(2)}_{k_2} - t^{(2)}\big)\right)(-1)$$

so that it gave me the actual equation back symbolically (if that happens to be the correct one).

Ideally I'd like to extend it to accept vector inputs to as in:

$$ f(x) = \sum^{K_2}_{k_2=1} c_{k_2} \exp\left(- \big\| a^{(2)} - t^{(2)}_{k_2}\big\|^2\right)$$

but one thing at a time.

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This is what I have tried so far. So before I try to me the upper limit of the sum symbolic, I am going to try to try it out a fixed $K_2$.

f[a_, t_, c_] = Sum[c[k2] * exp[a, t[k2]], {k2, 1, 3}]

which outputs:

which seems exactly correct. However, when I actually try to evaluate it and give it a vector it doesn't work as I expect:

instead of indexing at every location it just inserts the list for the variable, which isn't exactly what I had in mind. I was expecting something like:

$$ f = c_1 e^{-(a - t_1)^2} + c_2 e^{-(a - t_2)^2} + c_1 e^{-(a - t_1)^2} $$

$$ f = 5 e^{-(a - 1)^2} + 6 e^{-(a - 2)^2} + 7 e^{-(a - 3)^2} $$

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I also tried a simple example with K2 fixed and it didn't quite work:

why does it return zero :/

Answer 1

If I understand the question properly, you would like to differentiate the expression

f = Sum[c[k] Exp[-(a - t[k])^2], {k, kmax}]

with respect to t[n], where n is an integer between 1 and kmax, to obtain

(* 2 E^-(a - t[n])^2 c[n] (a - t[n]) *)

Almost certainly, a similar question has been raised before, but I cannot find it now. In any case, the obvious D[f, t[n]] with n a given integer, returns 0. For instance,

Assuming [kmax > 3, D[f, t[3]]]
(* 0 *)

And,

D[f, t[k]]
(* Sum[(2*c[k]*(a - t[k]))/E^(a - t[k])^2, {k, kmax}] *)

in effect treats all t[k] as the same variable. An inelegant and, from my perspective, unsatisfying approach does, however, produce the desired output.

D[f[[1]] /. k -> n, t[n]]    
(* 2 E^-(a - t[n])^2 c[n] (a - t[n]) *)

Of course, this works only because any given t[n] occurs only in one term of the Sum.

Answer 2

In M11.2, the direct approach now works:

f = Sum[c[k] Exp[-(a-t[k])^2], {k, kmax}];

Assuming[i ∈ Integers && 1<=i<=kmax, Simplify @ D[f, t[i]]]

2 E^-(a - t[i])^2 c[i] (a - t[i])