91Ó°ÊÓ

Partial derivative calculation is a technique used to find the derivative of a multi-variable function with respect to one variable while treating other variables as constants. This is essential for understanding how a function changes when altering just one of its variables. In this exercise, we are given the function \( f(x, y) = xye^{xy} \). To calculate its partial derivatives, we decide which variable to differentiate by and keep the other as a constant.

For the first partial derivative with respect to \( x \), known as \( f_x \), we differentiate \( xy e^{xy} \) treating \( y \) as a constant. This involves applying calculus rules, especially the product rule, to get:
* \( f_x = y e^{xy} + xy^2 e^{xy} \).

Similarly, to find the first partial derivative with respect to \( y \), \( f_y \), \( x \) is considered constant. We differentiate the function again to obtain:
* \( f_y = x e^{xy} + x^2y e^{xy} \).

Calculating these derivatives is foundational to further derive second-order derivatives and mixed partial derivatives.

The product rule is a fundamental concept in calculus used when differentiating products of two functions. If \( u(x) \) and \( v(x) \) are two separate functions of \( x \), their derivative is given by:

\[ \frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'. \]

This rule is crucial for solving our exercise, as seen in calculating the first partial derivatives. In this exercise, the function \( xy e^{xy} \) is viewed as a product of the two separate parts, \( xy \) and \( e^{xy} \).

By applying the product rule:
* When finding \( f_x \), we treat \( y \) as constant of \( x \) to differentiate, leading to the derivative of \( y e^{xy} + xy^2 e^{xy} \).
* For \( f_y \), the roles invert, with \( x \) being constant to \( y \), producing \( x e^{xy} + x^2y e^{xy} \).

The product rule is important for understanding how the combination of these terms results in the derivatives we derived.

Mixed partial derivatives refer to second-order derivatives where the partial derivatives are taken with respect to different variables sequentially. For a function of two variables, \( f(x, y) \), we often find derivatives like \( f_{xy} \) and \( f_{yx} \). According to Clairaut's theorem, if these mixed derivatives are continuous in a neighborhood of a point, they are equal: \( f_{xy} = f_{yx} \).

In this exercise, we computed:
* \( f_{xy} \) by differentiating \( f_x = y e^{xy} + xy^2 e^{xy} \) with respect to \( y \). * \( f_{yx} \) by differentiating \( f_y = x e^{xy} + x^2y e^{xy} \) with respect to \( x \).

Both derivatives resulted in \( e^{xy} + 2x y e^{xy} + x^2 y^2 e^{xy} \), confirming the equality \( f_{xy} = f_{yx} \). This principle helps verify the consistency and correctness of partial derivative calculations in multivariable calculus.