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?

Question

Ben 2013-8-9

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回答： Paul 2023-3-5

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))

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Answer 1

Walter Roberson 2013-8-9

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

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Dominic 2023-3-5

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.

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