How to symbolically differentiate a function with respect to a different function
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Let's say I have the following defined as symbolic functions:
phi(t),
phi_dot(t),
alpha(t),
alpha_dot(t),
and I have the equation
x = phi_dot*cos(alpha)
I want to take the derivative of x with respect to phi_dot;
y = diff(x, phi_dot)
but I cannot do so without getting the error that follows:
Error using mupadmex
Error in MuPAD command: Invalid variable. [stdlib::diff]
Error in sym/diff (line 44)
R = mupadmex('symobj::diff', S.s, x.s, int2str(n));
I can easily differentiate with respect to time, for example. However, I need the partial derivative of the function with respect to, for example, phi_dot. The output I expect from the above example would be:
y = cos(alpha)
Can anyone help me?
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Risposta accettata
Alexander
il 12 Giu 2012
It seems as if you cannot differentiate with respect to a symbolic function. Maybe it helps if you substitute phi_dot with a normal sym?
>> syms zz; y = diff(subs(x, phi_dot, zz), zz)
y(t) =
cos(alpha(t))
3 Commenti
Più risposte (2)
dmitry luchinskii
il 18 Gen 2018
Modificato: Walter Roberson
il 18 Gen 2018
why would you try Functional Derivative of Vector of Functionals https://www.mathworks.com/help/symbolic/functionalderivative.html
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Walter Roberson
il 12 Giu 2012
If phi_dot is not independent of cos(alpha) then diff(phi_dot * cos(alpha)) with respect to cos(alpha) is not going to be cos(alpha)
3 Commenti
Walter Roberson
il 12 Giu 2012
Let phi_dot(t) = cos(alpha(t)) . Now differentiate phi_dot(t) * cos(alpha(t)) with respect to phi_dot(t) . You are differentiating phi_dot(t)^2 with respect to phi_dot(t) so the answer must be 2 * phi_dot(t) which would be 2 * cos(alpha(t)), which is *not* the same as cos(alpha(t))
You can only differentiate phi_dot(t) * cos(alpha(t)) with respect to phi_dot(t) and get cos(alpha(t)) if you know that phi_dot(t) is independent of cos(alpha(t)) -- and you have not given us any reason to expect that the two are independent.
You are failing to apply the product rule, http://en.wikipedia.org/wiki/Product_rule
Unlike the situation where variables are assumed to be independent, functions are not assumed to be independent in calculus.
Alexander
il 13 Giu 2012
That sounds reasonable. That might be the reason why MuPAD refuses to differentiate w.r.t. phi_dot. I just assumed they are independent because that yielded the requested result.
You are right, Peter should double check that this is what he wants.
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