Directional Derivative

Definition

The directional derivative is a measure of the rate of change of a function in a specific direction at a given point. It represents the slope of the function in a particular direction, providing information about how the function is changing along that direction.

5 Must Know Facts For Your Next Test

1. The directional derivative of a function $f(x,y)$ at a point $(x_0, y_0)$ in the direction of a unit vector $\vec{u} = (u_1, u_2)$ is given by the formula: $\frac{\partial f}{\partial \vec{u}} = u_1 \frac{\partial f}{\partial x} + u_2 \frac{\partial f}{\partial y}$.
2. The directional derivative measures the rate of change of the function in the direction of the vector $\vec{u}$, and its magnitude is equal to the length of the gradient vector $\nabla f$ multiplied by the cosine of the angle between $\vec{u}$ and $\nabla f$.
3. The directional derivative is useful in understanding the behavior of a function near a point, as it provides information about the rate of change of the function in a specific direction.
4. The gradient vector $\nabla f$ points in the direction of the greatest rate of increase of the function, and the directional derivative in the direction of $\nabla f$ is equal to the magnitude of the gradient vector.
5. Directional derivatives are used in optimization problems to find the direction of steepest ascent or descent, which is important for finding local maxima and minima of a function.

Review Questions

• Explain how the directional derivative is related to partial derivatives and the gradient of a function.
  • The directional derivative of a function $f(x,y)$ at a point $(x_0, y_0)$ in the direction of a unit vector $\vec{u} = (u_1, u_2)$ is given by the formula $\frac{\partial f}{\partial \vec{u}} = u_1 \frac{\partial f}{\partial x} + u_2 \frac{\partial f}{\partial y}$. This formula shows that the directional derivative is a linear combination of the partial derivatives of the function, weighted by the components of the unit vector $\vec{u}$. The gradient vector $\nabla f$ points in the direction of the greatest rate of increase of the function, and the directional derivative in the direction of $\nabla f$ is equal to the magnitude of the gradient vector.
• Describe how the directional derivative is used in the context of tangent planes and linear approximations.
  • The directional derivative is closely related to the concept of the tangent plane to a surface at a point. The equation of the tangent plane to the surface $z = f(x,y)$ at the point $(x_0, y_0, f(x_0, y_0))$ is given by the linear approximation: $z \approx f(x_0, y_0) + \frac{\partial f}{\partial x}(x_0, y_0)(x-x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y-y_0)$. The coefficients of this linear approximation are the partial derivatives of the function, which are directly related to the directional derivative in any direction $\vec{u}$ through the formula $\frac{\partial f}{\partial \vec{u}} = u_1 \frac{\partial f}{\partial x} + u_2 \frac{\partial f}{\partial y}$. Therefore, the directional derivative provides important information about the behavior of the function near the point of interest, as captured by the tangent plane.
• Explain how directional derivatives are used in optimization problems to find local maxima and minima of a function.
  • Directional derivatives play a crucial role in optimization problems, as they provide information about the rate of change of a function in a specific direction. To find the local maxima and minima of a function $f(x,y)$, one can use the fact that the gradient vector $\nabla f$ points in the direction of the greatest rate of increase of the function. The directional derivative in the direction of the gradient vector $\nabla f$ is equal to the magnitude of the gradient vector, which represents the maximum rate of change of the function. By finding the points where the gradient vector is zero, one can identify the critical points of the function, which may correspond to local maxima or minima. The directional derivative can then be used to determine the nature of the critical points, as the sign of the directional derivative in certain directions can indicate whether the point is a local maximum, minimum, or saddle point.

Related terms

Partial Derivative: A partial derivative measures the rate of change of a function with respect to one of its variables, while treating the other variables as constants.

Gradient: The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function and has a magnitude equal to the maximum rate of change.

Tangent Plane: The tangent plane to a surface at a point is the plane that best approximates the surface near that point, and its equation is given by the linear approximation of the function.