# 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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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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### Accepted Answer

Walter Roberson
on 9 Aug 2013

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

##### 1 Comment

Dominic
on 5 Mar 2023

### More Answers (1)

Paul
on 5 Mar 2023

Check out functionalDerivative

syms y x(t) t

y = sin(x);

ydot = diff(y,t)

syms Dx(t)

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

functionalDerivative(ydot,Dx(t))

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