Final answer:
To find the directional derivative of a function at a point in a specific direction, calculate the dot product of the gradient of the function at that point with a unit vector in the given direction.
Step-by-step explanation:
The student is asking about the calculation of a directional derivative of a function f at a given point P in the direction of a vector v, which makes an angle θ with the x-axis.
To find the directional derivative, which measures the rate of change of the function f in the direction of v, you can use the gradient of f at point P, denoted as ∇f(P). The directional derivative, Dv(f), is then given by the dot product of the gradient and the unit vector in the direction of v.
Here are the general steps to compute the directional derivative:
Calculate the gradient of f, ∇f, at point P.
Normalize the vector v to get the unit vector in the direction of v, which is v/||v||.
Compute the dot product of ∇f(P) and the unit vector in the direction of v to find the directional derivative: Dv(f) = ∇f(P) · (v/||v||).
Note that the gradient is a vector that points in the direction of the steepest increase of the function, and its components are the partial derivatives of the function with respect to the variables.