Directional Derivative: The directional derivative measures the rate of change of a function in a particular direction. It tells us how the function's o...
Directional Derivative:
The directional derivative measures the rate of change of a function in a particular direction. It tells us how the function's output changes as we move a small amount in that direction.
Definition:
Let f(x, y) be a function defined on a region R in the plane. The directional derivative of f with respect to the vector v = (a, b) is defined as:
where (\partial f/\partial x) and (\partial f/\partial y) are the partial derivatives of f with respect to x and y, respectively.
Interpretation:
The directional derivative tells us that the rate of change of f in the direction of v is equal to the dot product of the gradient vector (\nabla f) and the vector v. (\nabla f) is a vector containing the partial derivatives of f.
Geometric Interpretation:
Consider a point P(x, y) in the domain of f. The directional derivative tells us the rate of change of f at P in the direction of v. It is equal to the magnitude of the gradient vector multiplied by the dot product of the gradient and the vector v.
Geometric Interpretation Continued:
If we think of the gradient vector (\nabla f) as a vector that points in the direction of the steepest ascent, then the directional derivative tells us how quickly the function is changing in that direction.
Examples:
If (f(x, y) = x^2 + y^2), then the directional derivative in the direction of the vector v = (1, 1) is 4, since (\partial^2 f/\partial x^2 = 2) and (\partial^2 f/\partial y^2 = 2).
If (f(x, y) = x\sin(y)), then the directional derivative in the direction of the vector v = (1, 0) is 0, since (\partial f/\partial x = \partial f/\partial y = \sin(y)).
The directional derivative tells us that the function (f(x, y) = x^3 + y^4) has a maximum at the point (1, 1)