Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the gradient \(\nabla f\). $$ f(x, y)=x e^{x y} $$

Step-by-Step Solution

Answer

\( \nabla f = \left( e^{xy}(1 + xy), x^2 e^{xy} \right) \).

Step 1: Identify the Function Components

The function given is \( f(x, y) = x e^{xy} \). It has two variables, \( x \) and \( y \). Our task is to find the gradient, which involves computing the partial derivatives with respect to each variable.

Step 2: Partial Derivative with Respect to \( x \)

To find \( \frac{\partial f}{\partial x} \), apply the product rule. The function can be split into two parts: \( u = x \) and \( v = e^{xy} \). The derivative using the product rule \( (uv)' = u'v + uv' \) becomes:- \( u' = 1 - \( v = e^{xy}, \frac{\partial v}{\partial x} = y e^{xy} \)\) Therefore, \( \frac{\partial f}{\partial x} = e^{xy} + x \, y \, e^{xy} = e^{xy}(1 + xy) \).

Step 3: Partial Derivative with Respect to \( y \)

Next, compute \( \frac{\partial f}{\partial y} \). Treat \( x \) as a constant. The expression \( x e^{xy} \) differentiates as follows:\( \frac{\partial}{\partial y} (x e^{xy}) = x \cdot \frac{\partial}{\partial y} (e^{xy}) = x \cdot x e^{xy} = x^2 e^{xy} \).So, \( \frac{\partial f}{\partial y} = x^2 e^{xy} \).

Step 4: Write the Gradient

The gradient \( abla f \) is a vector containing the partial derivatives found:\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = \left( e^{xy}(1 + xy), x^2 e^{xy} \right) \].

Key Concepts

Partial Derivatives

In the realm of multivariable calculus, partial derivatives play a critical role. They help determine how a function changes as one of its variables changes, while the other variables stay constant.
Partial derivatives are crucial for understanding the behavior of functions of multiple variables. Here are a few key points:

• A partial derivative with respect to a variable, say \( x \), involves treating all other variables, such as \( y \), as constants.
• For a function \( f(x, y) \), \( \frac{\partial f}{\partial x} \) indicates how \( f \) changes as \( x \) changes alone.
• In essence, partial derivatives allow us to "slice" through a multivariable function, viewing how it changes along each axis independently.

In the original solution, we find \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \), demonstrating these concepts effectively.

Product Rule

The product rule is a fundamental technique in calculus used when differentiating products of functions. When dealing with multivariable functions, the product rule comes in handy when a function can be expressed as a product of two simpler functions.
Here’s a quick rundown of the product rule:

• For two functions, \( u(x) \) and \( v(x) \), their product \( u(x)v(x) \) has a derivative given by: \( (uv)' = u'v + uv' \).
• In case of partial derivatives, treat each variable partially while applying the product rule on functions of multiple variables.

In the provided exercise, the function \( f(x, y) = x e^{xy} \) can be viewed as the product of \( u = x \) and \( v = e^{xy} \), making the product rule essential for computing \( \frac{\partial f}{\partial x} \).

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of multiple variables. This branch of calculus allows us to explore the vast landscapes within functions like \( f(x, y) = x e^{xy} \).
Unlike single-variable calculus:

• Multivariable calculus deals with functions that have two or more variables, offering new insights into those functions’ behaviors.
• It involves exploring how functions behave by examining cross-sections along various axes based on the partial derivatives.
• Concepts such as gradients, partial derivatives, and multiple integrations are cornerstone elements in understanding complex, multivariable systems.

This approach allows for the analysis of functions that depict surfaces, contours, or evaluate changes on a grid, which is crucial in physical sciences and engineering.

Vector Calculus

Vector calculus is a field that includes various operations pertinent to vector fields, such as gradient calculations. Gradient vectors give us directions of the steepest ascent in multivariable functions.
The gradient, denoted \( abla f \), serves as a vector containing all first order partial derivatives, showing the function's slopes:

• For the function \( f(x, y) = x e^{xy} \), the gradient is formed by placing \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) in a vector.
• The result, \( abla f = (e^{xy}(1 + xy), x^2 e^{xy}) \), indicates the direction in which \( f \) increases most rapidly.
• Understanding the gradient aids in areas like optimization, where knowing where changes happen fastest is invaluable.

By grasping vector calculus concepts, learners can efficiently navigate through complex surface topographies represented by functions with numerous variables.