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Answer 1 · 2017-09-12T18:26:23

Use implicit differentiation to solve for the derivative to get #dy/dx=(1-2x-ycos(xy))/(1+xcos(xy))#.

The given equation is #sin(xy)+x^2=x-y#.

We assume this equation defines #y# as a function of #x# (you can even write #f(x)# in place of #y# if you want).

Differentiating with respect to #x# and using this assumption along with the Chain Rule and Product Rule gives:

#cos(xy)*(1*y+x*dy/dx)+2x=1-dy/dx.#

Now multiply this out and rearrange to get the #dy/dx# terms on the left side:

#(xcos(xy)+1)*dy/dx=1-ycos(xy)-2x#.

Now divide to get the answer:

#dy/dx=(1-2x-ycos(xy))/(1+xcos(xy))#

BTW, the graph of the given curve is pretty "wild". It is shown below:

enter image source here