Question

Question #ef682

Answer 1

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

Explanation:

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: