How to implement a derivative of a symbolic function by a 'symfun' in Matlab?

Question

Here is my Matlab program: ``` syms x(t) t; f = x^2 + 5; ``` And the result of `f`: ``` f(t) = x(t)^2 + 5 ``` Both `f` and `x` are of class `symfun` in Matlab. I want the result of df/dx, which is equal to `2*x(t)`. I tried this in Matlab: ``` diff(f, x); ``` and got the following errors: ``` Error using mupadmex Error in MuPAD command: The variable is invalid. [stdlib::diff] Error in sym/diff (line 57) R = mupadmex('symobj::diff', S.s, x.s, int2str(n)); ``` How can df/dx be obtained in Matlab?

Accepted Answer

In newer versions of Matlab (I'm using R2014b) the error message is clearer:

Error using sym/diff (line 26) All arguments, except for the first one, must not be symbolic functions.

So, it looks like `sym/diff` cannot take a derivative with respect to what the documentation calls an abstract or arbitrary `symfun`, i.e., one without a definition. This limitation is not explicitly mentioned in the documentation, but the various input patterns allude to it.

Workaround 1: Simple functions, independent variable `t` only appears in differentiated symfun

I haven't found any workaround that is particularly elegant. If `t` only appears in your variable of interest (here `x(t)`), and not on its own or in other abstract symbolic functions, you can differentiate with respect to time `t` and then cancel out the extra terms. For example:

```
syms t x(t) y(t) z
f1 = x^2+5;
f2 = x^2+z^2+5;
f3 = x^2+y^2+z^2+5+t;
dxdt = diff(x,t);
df1 = diff(f1,t)/dxdt
df2 = diff(f2,t)/dxdt
df3 = diff(f3,t)/dxdt
```

which returns the requisite `2*x(t)` for the first two cases, but not the third, for the reasons stated above. You may need to apply `simplify` in some cases to fully divide out the `D(x)(t)` terms.

Workaround 2: More robust, but more complicated method

Using `subs` multiple times, you can replace the symbolic function in question with a standard symbolic variable, differentiate, and swap it back. For example:

```
syms t x(t) y(t) z
f3 = x^2+y^2+z^2+5+t;
xx = sym('x');
df3 = subs(diff(subs(f3,x,xx),xx),xx,x)
```

which also returns `2*x(t)` for the third case above. But I think this is kind of an ugly hack.

It is kind of ridiculous that Matlab can't do this – Mathematica has no such problem. It appears that MuPAD itself has this limitation. You might consider filing a feature request with The MathWorks.

How to implement a derivative of a symbolic function by a 'symfun' in Matlab?

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