[Tex/LaTex] A diagram about partial derivatives of f(x,y)

Question

How can I draw this image using TikZ?

Answer 1

Your question contains four pictures, out of which I am focusing on the lower two. Given that you want to vary the gray level of the plots, I'd like to recommend pgfplots where this kind of shading can be achieved with point meta. Like many other users, I am not too keen on punching in texts from screen shots, so I added some texts but you will find it easy to modify them to your needs.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{shadings}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}[bullet/.style={circle,fill,inner sep=1pt},
 declare function={f(\x,\y)=2-0.5*pow(\x-1.25,2)-0.5*pow(\y-1,2);}]
 \begin{axis}[view={150}{45},colormap/blackwhite,axis lines=middle,%
    zmax=2.2,zmin=0,xmin=-0.2,xmax=2.4,ymin=-0.2,ymax=2,%
    xlabel=$x$,ylabel=$y$,zlabel=$z$,
    xtick=\empty,ytick=\empty,ztick=\empty]
  \addplot3[surf,shader=interp,domain=0.6:2,domain y=0.5:1.2,opacity=0.7] 
   {f(x,y)};
  \addplot3[thick,domain=0.6:2,samples y=1]  ({x},1.2,{f(x,1.2)}); 
  \draw[dashed] (1.75,0,0) node[above left]{$x_0$} -- (1.75,1.2,0)
  node[bullet] (b1) {}  -- (0,1.2,0) node[above right]{$y_0$}
  (1.75,1.2,0) -- (1.75,1.2,{f(1.75,1.2)})node[bullet] {};
  \draw (1.75,1.2,{f(1.75,1.2)}) -- (0.75,1.2,{f(1.75,1.2)+0.5})
  coordinate[pos=0.5] (aux1);
  \draw[opacity=0.5,upper left=gray!80!black,upper right=gray!60,
lower left=gray!60,lower right=gray!80!black] (2,1.2,0) -- (0.6,1.2,0)
   -- (0.6,1.2,2.2) -- (2,1.2,2.2) -- cycle;
  \addplot3[surf,shader=interp,domain=0.6:2,domain y=1.2:1.9,opacity=0.7] 
   {f(x,y)};
 \end{axis}
 \draw (aux1) -- ++ (-1,1) node[above,align=center]{slope in $x$ direction\\
  $\partial_xf(x,y)|_{x=x_0,y=y_0}$};
 \node[anchor=north west] at (b1) {$(x_0,y_0)$}; 
 %
 \begin{axis}[xshift=6.5cm,view={150}{45},colormap/blackwhite,axis lines=middle,%
    zmax=2.2,zmin=0,xmin=-0.2,xmax=2.4,ymin=-0.2,ymax=2,%
    xlabel=$x$,ylabel=$y$,zlabel=$z$,
    xtick=\empty,ytick=\empty,ztick=\empty]
  \addplot3[surf,shader=interp,domain=0.6:1.75,domain y=0.5:1.9,opacity=0.7] 
   {f(x,y)};
   \addplot3[thick,domain=0.5:1.9,samples y=1]  (1.75,{x},{f(1.75,x)}); 
  \draw[dashed] (1.75,0,0) node[above left]{$x_0$} -- (1.75,1.2,0)
  node[bullet] (b2){}
  -- (0,1.2,0) node[above right]{$y_0$}
  (1.75,1.2,0) -- (1.75,1.2,{f(1.75,1.2)})node[bullet] {};
  \draw (1.75,1.2,{f(1.75,1.2)}) -- (1.75,0.2,{f(1.75,1.2)+0.2})
   coordinate[pos=0.5] (aux2);
  \draw[opacity=0.5,upper left=gray!80!black,upper right=gray!60,
lower left=gray!60,lower right=gray!80!black] (1.75,0.5,0) -- (1.75,1.9,0)
   -- (1.75,1.9,2.2) -- (1.75,0.5,2.2) -- cycle;
  \addplot3[surf,shader=interp,domain=1.75:2,domain y=0.5:1.9,opacity=0.7] 
   {f(x,y)};
 \end{axis}
 \draw (aux2) -- ++ (0.3,1) node[above,align=center]{slope in $y$ direction\\
  $\partial_yf(x,y)|_{x=x_0,y=y_0}$};
 \node[anchor=north east] at (b2) {$(x_0,y_0)$};
\end{tikzpicture}
\end{document}