91Ó°ÊÓ

In multivariable calculus, partial derivatives play a crucial role in understanding how a function changes with respect to one variable while keeping the others constant. For the function \( f(x, y) \), the partial derivatives are the rates at which \( f \) changes in the \( x \) and \( y \) directions.
To compute partial derivatives, follow these steps:

• For \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant and differentiate the function with respect to \( x \).
• For \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant and differentiate the function with respect to \( y \).

In our example, the partial derivative with respect to \( x \) is \( e^{xy} + xy e^{xy} \), and with respect to \( y \) is \( x^2 e^{xy} \). These calculations form the basis for building the gradient vector.

The product rule is essential for differentiating expressions where two functions are multiplied together. When you have a function \( u(x) \) multiplied by another function \( v(x) \), the product rule states that the derivative is:

• \( (uv)' = u'v + uv' \)

This rule is useful when dealing with expressions like \( x e^{xy} \) as seen in the exercise.
Here, when computing \( \frac{\partial f}{\partial x} \), consider \( x \) as \( u \) and \( e^{xy} \) as \( v \). Therefore, \( u' = 1 \) and \( v = e^{xy} \), applying the product rule gives us:
\[ \frac{\partial}{\partial x} [x e^{xy}] = e^{xy} + xy e^{xy} \]
Understanding how to use the product rule correctly is essential for differentiating complex functions effectively.

The chain rule is a technique for finding the derivative of a composite function. It is particularly useful when a function depends on another variable which in turn depends on a different variable. In simpler terms, if a function \( h(x) \) can be expressed as \( h(g(x)) \), the derivative using the chain rule is:

• \( h'(g(x)) \cdot g'(x) \)

In our exercise, the function \( e^{xy} \) involves an internal function \( xy \). When differentiating \( \frac{\partial}{\partial y} \), we apply the chain rule because \( e^{xy} \) is dependent on \( y \) via \( xy \).
So, we get:
\[ \frac{\partial}{\partial y} [x e^{xy}] = x \cdot \frac{\partial}{\partial y}[e^{xy}] = x \cdot xe^{xy} = x^2e^{xy} \]
The chain rule streamlines the process of differentiation when dealing with nested functions, making it a vital component of calculus.