Question #ef682

1 Answer
Sep 12, 2017

Use implicit differentiation to solve for the derivative to get dydx=12xycos(xy)1+xcos(xy).

Explanation:

The given equation is sin(xy)+x2=xy.

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)(1y+xdydx)+2x=1dydx.

Now multiply this out and rearrange to get the dydx terms on the left side:

(xcos(xy)+1)dydx=1ycos(xy)2x.

Now divide to get the answer:

dydx=12xycos(xy)1+xcos(xy)

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

enter image source here