Partial Derivatives of sin^2(x) + cos^2(y)

Separated variables make each partial a one term job

The function is a sum where the first term has only $x$ and the second only $y$. Differentiating with respect to $x$ kills $\cos^2 y$ outright, since a term with no $x$ in it is a constant.

The power rule supplies the factor $2\sin x$ and the chain rule supplies $\cos x$. The same structure on the other term gives $2\cos y$ times the derivative of cosine, which is where the minus sign comes from.

At $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ the partials are $\sin\frac{\pi}{2} = 1$ and $-\sin\frac{\pi}{2} = -1$, so the surface rises in $x$ and falls in $y$ at the same rate.

The mistake: reaching for sin^2 + cos^2 = 1

The Pythagorean identity needs the same angle in both terms. Here the arguments are $x$ and $y$, so $\sin^2 x + \cos^2 y$ is not $1$ and the function is not constant. At $x = 0$, $y = 0$ it equals $1$, but at $x = \frac{\pi}{2}$, $y = 0$ it equals $2$, and at $x = 0$, $y = \frac{\pi}{2}$ it equals $0$.

• Check the arguments match before applying any trigonometric identity.
• The values of this function fill the range from $0$ to $2$, so it is far from constant.
• If the identity did apply, $f$ would be constant and both partials would be zero everywhere, yet $\frac{\partial f}{\partial x} = 1$ at $\left(\frac{\pi}{4},\frac{\pi}{4}\right)$.

Critical points and the mixed partial

Both partials vanish when $\sin 2x = 0$ and $\sin 2y = 0$, that is when $x$ and $y$ are integer multiples of $\frac{\pi}{2}$. The critical points form a lattice across the plane.

Because no term contains both variables, the mixed partial is zero everywhere: $\frac{\partial^2 f}{\partial x\,\partial y} = 0$. That makes the second derivative test easy, since the discriminant is just $\frac{\partial^2 f}{\partial x^2}\cdot\frac{\partial^2 f}{\partial y^2}$ with $\frac{\partial^2 f}{\partial x^2} = 2\cos 2x$ and $\frac{\partial^2 f}{\partial y^2} = -2\cos 2y$.

At $(0,0)$ that gives $2 \cdot (-2) = -4 < 0$, so the origin is a saddle. Minimum in one direction, maximum in the other, which is exactly what a sum of two independent oscillations produces.