After taking the time derivative of a symbolic expression, how do you then differentiate that new expression with respect to another variable's time derivative?

I apologize for such a poorly worded question, but I'm not quite sure how else to word it.

You're given, x and y, both symbolic variables as a function of time. y is also a function of x. You determine the time derivative of y, ydot. xdot also appears in ydot because y is a function of x, and x is a function of time. How do you then differentiate ydot with respect to xdot?

Here's my code:

syms y x(t) t
y = sin(x);
ydot = diff(y,t);
diff(ydot,D(x)(t))

I've also tried this method:

syms y x(t) t
y = sin(x);
ydot = diff(y,t);
syms Dx
ydot = subs(ydot,sym('D(x)'),Dx);
diff(ydot,Dx)

But that gives me an output of:

ans(t) = 0

The answer should be cos(x(t))

Answer 1

[Walter Roberson]

No. MuPAD can only differentiate with respect to a variable, not with respect to a function.

[Dominic]

You have to create the general diff function for partial differentiation. You can perform this using the limit definition of the derivative with an h value as close to 0 as possible. You can program in the algebraic steps to simplification and then return the resultant expression. This will work for any type of function assuming the derivative is well defined.

Answer 2

[Paul]

Check out functionalDerivative

syms y x(t) t
y = sin(x);
ydot = diff(y,t)

ydot(t) = 

syms Dx(t)
ydot(t) = subs(ydot,diff(x,t),Dx(t))

ydot(t) = 

functionalDerivative(ydot,Dx(t))

ans(t) =