Indeed, most tools surprisingly lack this ability. I was shocked when I needed to break my calculations down to a piece-wise form when doing matrix calc with SymPy.

Mathematica can absolutely do all of this! D[ matrix ,x] ... or things like this:

f[x_,y_] := x^2 + Sin[y]

vars={x,y};

Table[ D[f[x,y],var1, var2], {var1,vars}, {var2,vars}] // MatrixForm

But how do you compute the derivative of x'Ax in Mathematica (x being a vector and A being a matrix)? What you have pointed out is only scalar derivatives, if I am not mistaken here.

Like this, perhaps?

A = {{1,2},{3,4}}

vec = {x^2, x^3}

D[vec.A.vec, x]

Or perhaps like this, again the table of derivatives:

xvec = {x1,x2}

Table[ D[xvec.A.xvec,x] ,{x,xvec}]

(all untested... one typo caught...)

Mathematica can definitely compute the derivatives if you fix the size of the matrix. This isn't very useful if you're trying to compute the derivative of an expression with arbitrary sized matrices.

You can do general matrices too, what do you have in mind?

aa[x_] = {{1, 2}, {3, 4}} x

bb[x_] = {x^2, x^3}

D[ a[x].b[x] , x, x] (* for any suitable tensors *)

% /. {a -> aa, b -> bb}

Added Link to Wolfram Alpha...

Note: This website presumes denominator layout, which is different from what is used in the guide.

Does the layout matter as long as you're consistent? Do the deep learning libraries presume that you're using a certain layout?

The deep learning libraries largely hide all the calculus - it's all automated.

Wow! Great little calculator. Thanks for pointing us at it.

You should add a link in the resources section

Done. Added a link.