Final answer:
The directional derivative of a function along a unit vector can be found using the dot product. The gradient represents the rate of change of the function in each direction, and the dot product gives us the component of that rate of change in the direction of the unit vector.
Step-by-step explanation:
The directional derivative of a function along a unit vector can be found using the dot product. Given a function and a unit vector, we can find the directional derivative by taking the dot product of the gradient of the function and the unit vector. The gradient represents the rate of change of the function in each direction, and the dot product gives us the component of that rate of change in the direction of the unit vector.
For example, if we have a function f(x, y) = x^2 + y^2 and we want to find the directional derivative at the point (2, 1) along the unit vector î + ĵ, we first calculate the gradient of f as ∇f = (2x, 2y), then evaluate it at the point (2, 1) to get ∇f(2, 1) = (4, 2). Finally, we take the dot product of ∇f(2, 1) and the unit vector î + ĵ to get the directional derivative.