Implicit Differentiation — Question 10

Question 10

Consider the real relation $$x^2+y^2=\sin(xy).$$

• Derive the formal relation obtained by implicit differentiation. State when division would be valid.

• Determine the real solution set and decide whether a tangent slope exists at $(0,0)$.

Question 10 - Solution

Formal differentiation. If a differentiable branch existed, it would obey

$$2x+2yy'=\cos(xy)(y+xy'),$$

so $(2y-x\cos(xy))y'=y\cos(xy)-2x$. Division would require a nonzero coefficient.

Analyze the real point set first. For every real $x,y$,

$$\sin(xy)\le|xy|\le\frac{x^2+y^2}{2}.$$

Thus the given equation implies

$$0\le x^2+y^2\le\frac{x^2+y^2}{2},$$

which forces $x=y=0$. The real solution set is the singleton $\boxed{\{(0,0)\}}$.

There is no curve branch through this isolated point and hence no tangent slope $dy/dx$ there. The formal expression $0/0$ alone would not establish this conclusion.