# 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?

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Ben on 9 Aug 2013
Answered: Paul on 5 Mar 2023
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

Walter Roberson on 9 Aug 2013
No. MuPAD can only differentiate with respect to a variable, not with respect to a function.
Dominic on 5 Mar 2023
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.

Paul on 5 Mar 2023
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) =