Question: Find a line that is tangent to both $\cos(xy) = 1+ \sin(y)$ and $y = x^2$
My first attempt is to differentiate both to get:
$$ (1) \ \frac{dy}{dx} = \frac{-y \sin(xy)}{\cos(y) + x \sin(xy)} \ (2) y' =2x$$
We need to find values where the tangent lines are the same so I set (1) to equal (2) and get:
$$2x = \frac{-y \sin(xy)}{\cos(y) + x \sin(xy)}$$
My first idea is to pick some value for $y$ so that I am able to solve for it, so I set $y = 0$. This produces the result $2x= 0$ and hence making the line of tangent $y = 0$.
However, I am wondering if there is a better way of getting a general solution without speculating on a good value for $y$. Thanks.
