Partial derivative of n-variables product

Question

I am new to Mathematica. I would like to calculate the partial derivative of the product, $\frac{\partial}{\partial x_{j}} y \prod _{i=1}^{n}x_{i}^3=\frac{3}{x_j} y \prod_{i=1}^nx_i^3$, where $1 < j < n$, and $\frac{\partial}{\partial y} y \prod _{i=1}^{n}x_{i}^3= \prod_{i=1}^nx_i^3$ .

I have try the code D[Product[y*Subscript[x, i]^3, {i, 1, n}],Subscript[x,j]] and Carl Woll's solution, but get 0. And the halirutan's method works but not neat one with piecewise production.

How to use Mathematica 12.0 to calculate derivatives and get a nice and exact result?

One can start here: https://mathematica.stackexchange.com/questions/16375/how-to-differentiate-formally — march, Sep 21 '21 at 15:12
Does this answer your question? Differentiate the product of some terms. In particular, check out my answer (which sadly has no votes). — Carl Woll, Sep 21 '21 at 16:11
I get Product[y*Subscript[x, i]^3, {i, 1, n}]*Sum[(3*KroneckerDelta[i, j])/Subscript[x, i], {i, 1, n}] in M12.0.1 using my answer. Have you tried from a fresh kernel? Also, it might work better if you use x[i] instead of Subscript[x, i]. — Carl Woll, Sep 22 '21 at 16:26

score 2 · Answer 1 · answered Sep 21 '21 at 14:08

2

Without knowledge of n,j the expression cannot be evaluated.

For example

D[Product[Subscript[x, i]^3, {i, 1, 5}], Subscript[x, 3]]

evaluates as expected

answered Sep 21 '21 at 14:08

Ulrich Neumann

53,729
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How can I evaluate it without specifying j=3？ – XJ.C Sep 21 '21 at 14:39

Partial derivative of n-variables product

1 Answers1