Directional Derivative
Text The directional derivative for a function f(x,y) in the direction θ is denoted ∇_{θ}f(x,y) and is the derivative of the height function z(t) in that direction.
Consider a point P = (x_{0},y_{0}) in the domain of f(x,y). The line in the domain corresponding to the direction θ will have the parametric form (x_{0} + t cos θ, y_{0} + t sin θ) and the height function z(t) associated with the slice curve along this line will have the form z(t) = f(x_{0} + t cos θ, y_{0} + t sin θ).
The directional derivative ∇_{θ}f(x_{0},y_{0}) at the point P will be the derivative evaluated at t = 0 of the height fuction z(t)
∇_{θ}f(x_{0},y_{0}) = z'(t) ^{t=0} = ∂/∂t (f(x_{0}+t*cos(θ),y_{0}+t*sin(θ)))^{t=0}.
Note that the partial derivatives f_{x} and f_{y} 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.
Demos
Tangent Line to Slice Curves
 
Recall from Lab 2 that the partial derivatives of a function f(x,y) at a point (x_{0},y_{0}) can be found by looking the x and yslice curves of the function graph and then calculating the slopes of the tangent lines to these curves at (x_{0},y_{0},f(x_{0},y_{0}). For the directional derivative, instead of slicing along the positive x and ydirections, we slice the graph along a direction (cos θ, sin θ). The slope of the tangent line to the slice curve at (x_{0},y_{0}, f(x_{0},y_{0}) is the directional derivative. The demo allows you to choose a point (x_{0},y_{0}) in the domain as well as a direction (cos θ, sin θ) using the two orange hotspots. The graph is shown in a separate window with the corresponding slice curve and slicing plane. The third window shows the slice curve in its plane along with the tangent line.

Exercises 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.
