Partial derivatives
At the saddle's center $(0, 0)$ the map shows higher ground east and west, lower ground north and south. What are the two slice slopes $f_x$ and $f_y$ there?
At $(1, 0.5)$ the east arrow is twice the north arrow: stepping east, the bowl climbs at rate $2$; stepping north, at rate $1$. Cross to $x < 0$ and the $x$-arrow points west — that slice now descends. At the saddle's center both arrows vanish and the stats read $0.000$ and $0.000$, yet the map around the probe is anything but featureless.
A partial derivative is an ordinary derivative applied to a slice. Freeze $y$ at $0.5$ and the bowl collapses to a one-variable function $g(x) = x^2 + 0.25$, whose derivative $g'(x) = 2x$ is the partial $f_x$. Differentiate treating $x$ as the constant instead and $f_y = 2y$. At $(1, 0.5)$ that gives $f_x = 2$ and $f_y = 1$ — the two numbers on the stats. The dashed crosshair through the probe shows the two slices being measured.
PREREQUISITEEach arrow is a tangent slope from single-variable calculus — one slice at a time.The saddle settles the prediction: $f_x = 2x$ and $f_y = -2y$ both vanish at the center, exactly as at a bowl's bottom — but here the $x$-slice curves up while the $y$-slice curves down. Two flat slices do not make a peak or a pit. Points where both partials vanish are called critical points; deciding which kind takes more than the two slopes.
On the bowl at $(1, 0.5)$ the stats read $f_x = 2.000$. What does that number mean?
For the bowl, $f_x = 2x$ and $f_y = 2y$. What do the stats read at $(-1,\ 1.5)$?
Transfer: $g(x, y) = 3x + x^2 y$. What is $g_y$, the partial with respect to $y$?
A partial derivative freezes every variable but one and differentiates the slice that remains. $f_x$ and $f_y$ are the surface's slopes along the two axis directions — and both can vanish at a point that is neither a peak nor a pit.