Partial Derivatives of Functions (continued)

Physical Interpretation of the partial derivative ∂f(x,y)/ ∂y

Consider a surface S given by z = f(x,y). The intersection of the plane, PL, given

by x = constant with the surface, S, defines the curve of intersection, C. Let the

tangent line to C at point P with coordinates (a, b, c) be T.

Then the slope of the line tangent to the curve C at the point (a,b,c) in the y-direction is:

f_y(x,y) = ∂f(x,y)/ ∂y evaluated at the point P (a,b,c)

Similarly one can imagine a plane, y = constant, intersecting the surface z

along a curve D (not shown).

The slope of the line tangent to the curve D at the point (a,b,c) in the x-direction is:

f_x(x,y) = ∂f(x,y)/ ∂x evaluated at the point P (a,b,c)

Click here for an example. Click here for discussion of linear approximation.

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