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The quotient rule is a tool used in calculus to differentiate functions that are the ratio of two differentiable functions. In this scenario, we are working with a function in the form of a fraction, with the numerator and denominator being functions of variables. The quotient rule is expressed as:

• If you have a function \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) \) is \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).

In our example, the function \( z = \frac{x^2 y^2}{x^2 + y^2} \) is differentiated using the quotient rule. Here, \( u(x, y) = x^2y^2 \) and \( v(x, y) = x^2 + y^2 \). Using the quotient rule assists in breaking down the complex differentiable expression into manageable parts, where the derivatives of both the numerator and the denominator are calculated separately and then combined according to the rule. This is particularly useful for multi-variable functions where each variable’s contribution needs to be considered.

Partial derivatives are an essential concept in the study of multivariable calculus. When a function depends on more than one variable, partial derivatives are used to explore the rate of change with respect to one of those variables while keeping the others constant.

• It is denoted by the symbol \( \frac{\partial}{\partial x} \) or \( \partial_x \) for a variable \( x \).

In the given problem, we seek the partial derivatives of \( z \) with respect to \( x \) and \( y \). This involves treating one variable as a constant while differentiating with the other, using the quotient rule to simplify the process for a fraction-like function. For example, for \( \frac{\partial z}{\partial x} \), we apply the quotient rule by taking the derivative of \( x^2y^2 \) with respect to \( x \), which results in \( 2xy^2 \), while considering the \( y \) component to be constant. Similarly, for \( \frac{\partial z}{\partial y} \), the differentiation considers \( x \) as constant, resulting in \( 2x^2y \). This calculation provides insight into how the function changes as each variable independently varies.

A differential equation is an equation involving functional derivatives, which represent rates of change. Solving a differential equation provides a function or set of functions that satisfies the relationship dictated by the equation. In this example, our task was to demonstrate that a given function satisfies a specific differential equation. By substituting the partial derivatives calculated with the quotient rule back into the equation \( x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 2z \), we were able to verify the solution. The equation simplifies to confirm that our original function \( z = \frac{x^2 y^2}{x^2 + y^2} \) indeed satisfies this relationship.

• This involves not just computing derivatives but combining them to match the pattern needed on both sides of the equation to verify the solution.

The beauty of solving differential equations lies in how derivatives are used to piece together a bigger picture, answering how a system behaves under changing conditions. In this case, we verified that the function is consistent with the differential equation related to its structure and interaction of its variables.