How can I differentiate an abstract symbolic function?

Hi, I would like to differentiate an abstract symbolic function in MATLAB.

To make sure we're all on the same page, let's say I have a function of two variables where f is any function of one variable that is differentiable across its domain. Using partial differentiation, we should expect to see:

and

I've tried to use syms to observe this behaviour in MATLAB:

syms x y f(k) g(x,y)

g(x,y) = f(x - 3*y);
g_x = diff(g,x);
g_y = diff(g,y);

In effect, we find that

g_y = -3*D(f)(x - 3*y)

Which translates to after using the latex function.

But we also find that

g_x = diff(f(x - 3*y), x)

which translates to .

I have tried to use the expand and simplify functions on this result, but they don't seem to transform it into the expected . Is there a way for me to do this?

Note that if the coefficient of x was anything other than 1 or 0, then we have

g_x = c*D(f)(c*x - 3*y)

Answer 1

[Star Strider]

I am not certrain that I understand what you want to do.

See if the diff documentation section on Differentiation With Respect to Function and Derivative will do what you want. It requires a slightly different syntax.

.

[c2332]

I was trying to represent as since subtracting the two representations will not equal zero.

"I have tried to use the expand and simplify functions on this result, but they don't seem to transform it into the expected . Is there a way for me to do this?"

[Star Strider]

I do not understand what you want to do. It appears that you want a specific format, and that format may not be available, unless you create it yourself.

.

Answer 2

[Paul]

When I run this code (2019a) on my computer, I get:

syms x y f(k) g(x,y)
g(x,y) = f(x - 3*y);
g_x = diff(g,x)
g_y = diff(g,y)

g_x(x, y) =
 
D(f)(x - 3*y)
 
 
g_y(x, y) =
 
-3*D(f)(x - 3*y)

Is that not the form of the expected result? I'm confused if what you're asking about is the form of the result as sym objects, or the form of the typeset output from the Live Editor. If the latter, in my local version of Matlab I do get , although that desn't seem to be the case in 2021a, at least as run here:

g_x

g_x(x, y) = 

[c2332]

Your first block of code does provide the expected result, but the represenation at the end is not what I'm looking for.

So from what I can see, it's not possible to represent as in the later versions of MATLAB.

[Paul]

So this is really just an issue of how Live Script is formatting the output for viewing?

[c2332]

It's not just that issue. If I try performing

3*diff(f(x-3*y),x) + diff(f(x-3*y), y)

I should expect to get zero, but I get the following instead:

3*diff(f(x - 3*y), x) - 3*D(f)(x - 3*y)

[Paul]

Something changed since 2019a. I just ran in 2020b and got:

g_x(x, y) =
 
diff(f(x - 3*y), x)