What is the geometric difference between partial derivative and ordinary derivative? Ordinary derivative of a function let $~y=f(x)~$ represents the slope of the tangent drawn at the point $~x~$ to the curve $~f(x)~$. But, what about partial derivative ? Please, attach images if possible for understanding.
 A: A partial derivative gives us the slope of tangent line to a surface at a cross-section.

$~\frac{∂}{∂x}f(x,y)~$ means the rate at which $~f(x,y)~$ will change with respect to $~x~$ if $~y~$ is kept constant.
i.e.,  if you take a cross section of the surface through a specific value of $~y~$ then you will get the slope of the tangent to the surface at that cross section. This slope varies as $~x~$ varies. If you move one unit along the $~x$-direction at a cross section, then you will also move $~\frac{∂f}{∂x}~$units along $~z$-axis.

Similarly, $~\frac{∂}{∂y}f(x,y)~$ means the rate at which $~f(x,y)~$ will change with respect to $~y~$ if $~x~$ is kept constant.
For example, in figure 114.7, $~y~$ is held constant at $~y = b~$ and the red curve describes the curve $~x = f(x,b)~$ on the surface. The value of $~\frac{∂}{∂x}f(x,y)~$ is the slope of the tangent line to the curve at the point $~x = a~$ and $~y = b~$. Similarly for $~\frac{∂}{∂y}f(x,y)~$.

(http://www.tectechnicsclassroom.tectechnics.com/s7p5a51/geometric-interpretation-of-partial-derivatives.php)
