This applet illustrates the concept of partial derivative.

Check the box "y constant" to plot the plane that passes through the point with constant, and observe the curve that results from the intersection of the plane and the surface.
The partial derivative of with respecto to measures the instantaneous rate of change of when changes but keeps constant.
Geometrically is the slope of the tangent line to the curve that results from the intersection of the plane and the surface.

Check the box "x constant" to plot the plane that passes through the point with constant, and observe the curve that results from the intersection of the plane and the surface.
The partial derivative of with respecto to measures the instantaneous rate of change of when changes but keeps constant.
Geometrically is the slope of the tangent line to the curve that results from the intersection of the plane and the surface.

Move the x and y sliders to change the point and observe how the partial derviatives change.

Enter a new function and repeat the previous steps.