Labware - MA35 Multivariable Calculus - Two Variable Calculus



Directional Derivative


The directional derivative for a function f(x,y) in the direction θ is denoted θf(x,y) and is the derivative of the height function z(t) in that direction.

Consider a point P = (x0,y0) in the domain of f(x,y). The line in the domain corresponding to the direction θ will have the parametric form (x0 + t cos θ, y0 + t sin θ) and the height function z(t) associated with the slice curve along this line will have the form z(t) = f(x0 + t cos θ, y0 + t sin θ).

The directional derivative θf(x0,y0) at the point P will be the derivative evaluated at t = 0 of the height fuction z(t) θf(x0,y0) = z'(t) |t=0 =

∂/∂t (f(x0+t*cos(θ),y0+t*sin(θ)))|t=0.

Note that the partial derivatives fx and fy are just the directional derivatives at θ = 0 and θ = π/2, respectively.

Also note that the directional derivative is a function of the direction variable θ; the variable t is only used as a parameter.



  • Find the value of θ for the maximum directional derivative of any slice curve. What θ gives the minimum directional derivative for this same slice curve? How are these values related? Try this for several different slice curves.