$\begingroup$The following sets do not match.
D[(a*x+b*y[x]-x*(a*x+b*y[x])-y*(c*x + d*y[x]))*x, {x,1}] (correct answer) (a x + b y - x (a x + b y) - y (c x + d y) ) + (a - b - 2 a x - b y + b x - c y + c x + 2 d y) x (wolfram output) => Different from the above. Is this something that happens in general?
$\endgroup$41 Answer
$\begingroup$Here is an example of $$ \frac{\partial^2}{\partial x^2} \left[\sin(xy)\right] = -y^2 \sin(xy) $$ via Wolfram Alpha, so partial differentiation does work.
Here is another one related to your query, just not that complex: $$ \frac{\partial^2}{\partial x^2} \left[(ax+by(x))\cdot x\right] = 2ax + by(x) + bxy'(x) $$ so seems to work as well.
$\endgroup$
