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## The Geometry of Partial Derivatives
Information about the partial derivatives of a function $z = f(x,\,y)$ can be detected also from a contour map of $f$. Indeed, as one knows from using contour maps to learn whether a path on a mountain is going up or down, or how steep it is, so the sign of the partial derivatives of $z = f(x,y)$ and relative size can be read off from the contour map of $f$.
Solution 1 (continued): It appears that $\displaystyle
f_x\bigl|_{S} \,<\,0,\quad f_y\bigl|_{S} \,=\, 0\,.$
Solution 2: It appears that $\quad
f_x\big|_R>0,\quad f_y\big|_R<0,\quad f_x\big|_S=0,\quad
f_y\big|_S=0, \quad f_x\big|_Q>0,\quad f_y\big|_Q=0,\quad$ $
f_x\big|_P=f_y\big|_P=0,\quad f_x\big|_T<0,\quad f_y\big|_T=0$All the same ideas carry over in exactly the same way to
functions $w = f(x,\,y,\,z)$ of three or more variables - just
don't expect lots of pictures!! The partial derivative $f_z$, for
instance, is simply the derivative of $f(x,\,y,\,z)$ with respect
to $z$, keeping |