Chain rule and Subs

[Miguel Angel Salazar de Troya]

Hello

I have this derivative of a function that has an argument which is also a function

# Declaration

x = symbols('x')

U, Theta = symbols('U Theta', cls=Function)

# Function to derive

obj_function = Theta(U(x),x)

# Derivative

gradient_obj_function = obj_function.diff(x)

And this is the output I obtain

Derivative(Theta(U(x), x), x) + Derivative(U(x), x)*Subs(Derivative(Theta(_xi_1, x), _xi_1), (_xi_1,), (U(x),))

How can I replace Subs(Derivative(Theta(_xi_1, x), _xi_1), (_xi_1,), (U(x),)) with Derivative(Theta(U(x), x), U(x)) ?

I found this post, https://groups.google.com/forum/#!searchin/sympy/"chain$20rule"/sympy/OfF5PklJtmY/t1FPWt3n8xIJ, but it was not clear to me how this has to be done. 

Thanks

Miguel

[Isuru Fernando]

Hi,

I'm not sure how to do the substitution, but why do you expect  obj_function.diff(x) to be 

Derivative(Theta(U(x), x), x) + Derivative(U(x), x)* Derivative(Theta(U(x), x), U(x)) ?

I think the answer you are getting from sympy is not correct.
obj_function.diff(x) should be the same as Derivative(Theta(U(x), x), x), but that is not what sympy is giving. 

See, https://github.com/sympy/sympy/issues/8510

CSymPy is giving, 
Derivative(U(x), x)*Subs(Derivative(Theta(_x, x), _x), (_x), (U(x))) + Subs(Derivative(Theta(U(x), _x), _x), (_x), (x)) 

which I think should be the correct answer. 

Here what  Subs(Derivative(Theta(U(x), _x), _x), (_x), (x)) means is partial differentiate Theta with respect to the second variable and replace the first variable with U(x) and the second with x. 

(D[2](Theta)(U(x), x) is the equivalent in Maple syntax or Derivative[0, 1][Theta][U[x], x] in Mathematica syntax)

Isuru Fernando

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[Miguel Angel Salazar de Troya]

In the end, what I would like to have is this in Latex Code

\frac{\partial \Theta(U(x),x)}{\partial U(x)} \frac{U(x))}{\partial x} + \frac{\partial \Theta(U(x),x)}{\partial x}

and I want to obtain that just from the expression

# Declaration

x = symbols('x')

U, Theta = symbols('U Theta', cls=Function)

# Function to derive

obj_function = Theta(U(x),x)

and derive that w.r.t. "x" which is what mathematically makes sense to me.