14.3.1-241015 Partial Derivative Geometrically

Rotate the "xyz 3D-space" to view the yz-plane (so looking down the x-axis).

The LINE in the YZ-plane through the purple point (y,z)=(2,2) and having slope m=-1

has the equation (z-2) = (-1) (y-2), which simplifies

to z = 4 - y.

Let stick the line z= 4-y into the plane x=1 ... below, think of the "t" as our "y"

Lessons learned.

The partial derivative of z=f(x,y) w.r.t. y [so think of x as fixed], evaluated at the point (x_0, y_0), is the slope of the tangent line in the plane x=x_0 to the curve which is the intersection of graph of z=f(x,y) and the plane x=x_0.

See def. of Partial Derivative Remark part 1.

Set to TRANSLUCENT surface (wrench icon).

Rotate to our usual 3D view.

Similarly.

The partial derivative of z=f(x,y) w.r.t. x [so think of y as fixed], evaluated at the point (x_0, y_0), is the slope of the tangent line in the plane y=y_0 to the curve which is the intersection of graph of z=f(x,y) and the plane y=y_0.

All Done .