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Partial derivatives are a key concept in calculus when dealing with functions of multiple variables. They represent the rate of change of a function with respect to one variable, while keeping the other variables constant.
For example, given a function like our exercise function, \( f(x, y)=\frac{y}{x+y} \), we have two partial derivatives: \( f_x \) and \( f_y \), indicating derivatives with respect to \( x \) and \( y \) respectively.

• \( f_x \) tells us how \( f \) changes as \( x \) changes, with \( y \) being constant.
• \( f_y \) tells us how \( f \) changes as \( y \) changes, with \( x \) being constant.

Calculating these derivatives involves using rules like the chain rule and product or quotient rules, which are crucial for accurately determining how functions behave in a multidimensional space.

The quotient rule is an essential tool in calculus for differentiating functions that are expressed as a division of two other functions. If you have a function like \( f(x) = \frac{u(x)}{v(x)} \), the quotient rule states:\[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \]This rule is vital because it allows us to compute the derivative of a function efficiently when it is structured as a fraction.
In our exercise, we used the quotient rule to find the partial derivatives of the function \( f(x, y) = \frac{y}{x+y} \). Differentiating with respect to \( x \) and \( y \), we set up and solved for \( f_x \) and \( f_y \).
This calculation is crucial as it simplifies the process of finding both first-order and second-order partial derivatives, allowing us to handle complex functions with ease.

Clairaut's Theorem is a fascinating result in the study of partial derivatives, especially when exploring mixed partial derivatives. The theorem states that if the mixed partial derivatives of a function are continuous, then they are equal.
For a function \( f(x, y) \), this means:

• \( f_{xy} \) and \( f_{yx} \) are equal.

In our exercise, after calculating \( f_{xy} \) and \( f_{yx} \), we found that they indeed match, demonstrating Clairaut's theorem's applicability and significance. This theorem simplifies many problems involving multivariable functions, providing checks and balances when deriving mixed partial derivatives. Its utility becomes apparent in fields like physics and engineering where such functions are common.

Mixed partial derivatives are derivatives taken with respect to more than one variable. For instance, given a function \( f(x, y) \), the mixed derivatives \( f_{xy} \) and \( f_{yx} \) involve differentiating with respect to \( x \) and then \( y \), or vice versa.

• \( f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) \)
• \( f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) \)

These derivatives give insight into the interaction between variables in a function, showing how a change in one variable affects the rate of change with respect to another. This is especially important in contexts where variables do not operate independently.
In our exercise, finding \( f_{xy} \) and \( f_{yx} \) showed how the interplay between \( x \) and \( y \) affects the function \( f(x, y) = \frac{y}{x+y} \). Calculating mixed derivatives is useful for understanding critical points and behavior of multivariable functions in a variety of scientific and engineering applications.