Problem 295 - Calculus 3

Question

For the following exercises, find the derivative of the function. \(f(x, y)=e^{x y}\) at point \((6,7)\) in the direction the function increases most rapidly

Step-by-Step Solution

Answer

The derivative in the direction of steepest ascent at point (6, 7) is \( e^{42} \sqrt{85} \).

Step 1: Find the Gradient of the Function

The gradient of a function, denoted by \( abla f \), represents the direction of the steepest ascent of the function at any given point. For the function \( f(x, y) = e^{xy} \), compute the partial derivatives with respect to \( x \) and \( y \). \( \frac{\partial f}{\partial x} = y e^{xy} \) and \( \frac{\partial f}{\partial y} = x e^{xy} \). Therefore, the gradient is \( abla f = (y e^{xy}, x e^{xy}) \).

Step 2: Evaluate the Gradient at the Given Point

Substitute the point \((6, 7)\) into the gradient. \( \frac{\partial f}{\partial x} = 7 e^{42} \) and \( \frac{\partial f}{\partial y} = 6 e^{42} \). Thus, the gradient at the point \((6, 7)\) is \( abla f = (7 e^{42}, 6 e^{42}) \).

Step 3: Determine the Direction of Maximum Increase

The direction of maximum increase of the function is given by the vector pointing in the direction of the gradient, \( abla f \). Thus, the direction of steepest ascent at point \((6, 7)\) is along the vector \((7 e^{42}, 6 e^{42})\).

Step 4: Find the Magnitude of the Gradient Vector

The magnitude of the gradient \( abla f \) is found using the formula for Euclidean norm: \( ||abla f|| = \sqrt{(7 e^{42})^2 + (6 e^{42})^2} = e^{42} \sqrt{49 + 36} = e^{42} \sqrt{85} \).

Step 5: Write the Directional Derivative

The directional derivative along the gradient at point \((6, 7)\) in the direction of steepest ascent is equal to the magnitude of \( abla f \), which is \( e^{42} \sqrt{85} \). This value represents the rate of change of the function at the point in the direction where it increases most rapidly.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, used to understand how a function changes as each variable is varied independently. Unlike ordinary derivatives, which measure change along a single dimension, partial derivatives look at change along multiple dimensions. For a function like \( f(x, y) = e^{xy} \), we want to see how the function behaves when only \( x \) or \( y \) changes while the other stays constant.

• The partial derivative with respect to \( x \), \( \frac{\partial f}{\partial x} = y e^{xy} \), tells us how much \( f \) will change per unit change in \( x \), holding \( y \) constant.


• Similarly, \( \frac{\partial f}{\partial y} = x e^{xy} \) measures the change in \( f \) when \( y \) changes, with \( x \) held constant.

These derivatives help form the building blocks of the gradient and offer insight into how a function's surface evolves in its input space.

Directional Derivative

The directional derivative is a crucial idea when dealing with functions of several variables. It extends the concept of the derivative to directions other than parallel to the coordinate axes. In simpler terms, it tells us the rate of change of a function in any specified direction.

To find the directional derivative at a point, we rely on the gradient. The formula is:

• The directional derivative of a function \( f \) at a point \( (x_0, y_0) \) in the direction of a unit vector \( \mathbf{u} = (u_1, u_2) \) is given by: \( D_\mathbf{u}f(x_0, y_0) = abla f(x_0, y_0) \cdot \mathbf{u} \).


• This dot product results in a scalar value, representing how much \( f \) changes in that specific direction.

In our exercise, the function's change is found in the direction of the gradient itself, where change is most significant. This measurement aligns with the steepest ascent, simplifying the directional derivative to just the gradient's magnitude.

Steepest Ascent

Steepest ascent is the direction where a function increases the quickest. When we explore a function's behavior, we often look for paths of steepest ascent to describe how a function optimizes or climbs. This is inherently connected to the gradient of the function, \( abla f \).

• The vector of the gradient \( abla f = (y e^{xy}, x e^{xy}) \) points directly in the direction of the steepest ascent.


• At any point, its magnitude, given as \( e^{42} \sqrt{85} \) in the solution, tells us the rate of change in that direction.

By evaluating the gradient at a particular point, the path of steepest ascent is clear, pointing exactly along the direction of the gradient vector. Understanding this helps in optimizing functions and finding maximum values in various dimensions.