Problem 49 - Calculus an Applied Approach

Question

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\frac{e^{2 x y}}{4 x} $$

Step-by-Step Solution

Answer

The four second partial derivatives are \( \frac{\partial^2 z}{\partial x^2} = \frac{e^{2xy}}{x^3} - \frac{y e^{2xy}}{x^2} + \frac{y^2 e^{2xy}}{2x} \), \( \frac{\partial^2 z}{\partial x \partial y} = \frac{e^{2xy}}{x^2} \), \( \frac{\partial^2 z}{\partial y^2} = e^{2xy} \), \( \frac{\partial^2 z}{\partial y \partial x} = \frac{e^{2xy}}{2x} \). The second mixed partials are equal, \( \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} = \frac{e^{2xy}}{2x} \), confirming Schwarz's theorem.

Step 1: Find first partial derivatives

First, we need to compute the first order partial derivatives, denoted \( \partial z/\partial x \) and \( \partial z/\partial y \). Let us use the quotient rule for differentiation in terms of x, and the chain rule for differentiation in terms of y: \[ \frac{\partial z}{\partial x} = \frac{-e^{2xy}}{4x^2} + \frac{y e^{2xy}}{2x} \], \[ \frac{\partial z}{\partial y} = \frac{x e^{2xy}}{2x} = \frac{e^{2xy}}{2} \]

Step 2: Compute second partial derivatives

Using the results from Step 1, we can compute the second order partial derivatives. Simply differentiate \(\partial z/ \partial x\) with respect to x, \(\partial z/ \partial x\) with respect to y, \(\partial z/ \partial y\) with respect to y, and \(\partial z/ \partial y\) with respect to x: \[ \frac{\partial^2 z}{\partial x^2} = \frac{e^{2xy}}{x^3} - \frac{y e^{2xy}}{x^2} + \frac{y^2 e^{2xy}}{2x} \], \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{e^{2xy}}{x^2} - \frac{y e^{2xy}}{2x} = \frac{e^{2xy}}{x^2} \], \[ \frac{\partial^2 z}{\partial y^2} = e^{2xy} \], \[ \frac{\partial^2 z}{\partial y \partial x} = \frac{e^{2xy}}{2x} - \frac{y e^{2xy}}{4x^2} = \frac{e^{2xy}}{2x} \]

Step 3: Verify the Equality of Second Mixed Partials

According to Schwarzs's theorem, the second mixed partials \( \frac{\partial^2 z}{\partial x \partial y} \) and \( \frac{\partial^2 z}{\partial y \partial x} \) should be equal. Comparing our results from Step 2, we can see both of them are indeed \( \frac{e^{2xy}}{2x} \), so they indeed are equal, thus verifying the theorem.

Key Concepts

Quotient Rule

The Quotient Rule is a method used in calculus to differentiate functions that are divided by one another. This rule is essential for finding the derivative of a quotient, which is a function divided by another function. Let's break it down:

• Suppose you have a function \( z = \frac{u(x,y)}{v(x,y)} \).
• The quotient rule states that the derivative \( \frac{d}{dx}\left(\frac{u}{v}\right) \) is given by the formula: \( \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \).

In this exercise, to find the partial derivative with respect to \( x \), the Quotient Rule was used. The numerator of the derivative involves multiplying the derivative of the numerator, \( e^{2xy} \), while applying the rule, with the partial derivative of the exponent. This ensures we accurately consider how both the numerator and denominator of our fraction are changing with respect to \( x \).

Chain Rule

The Chain Rule is a fundamental technique in calculus used when differentiating composite functions.

• It helps to find the derivative of functions that are nested within other functions.
• If \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is \( f'(g(x)) \cdot g'(x) \).

In the context of this exercise, the Chain Rule is applied while finding the partial derivative with respect to \( y \). The function \( e^{2xy} \) requires using the Chain Rule due to the exponent \( 2xy \). This means taking the derivative of the outer function \( e^u \) (which is simply \( e^u \)), and multiplying it by the derivative of the inner function \( 2xy \) with respect to \( y \), which is \( 2x \).

Mixed Partials Equality

Mixed partial derivatives refer to derivatives taken with respect to different variables in a function of several variables. An important property of mixed partials is that, under certain conditions, they can be equal.

• According to Schwarz's theorem (also known as Clairaut's theorem), if \( f(x, y) \) is continuous and all the second partial derivatives exist, then \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \).
• This symmetry in partial derivatives simplifies multi-variable calculus problems.

In our problem, the equality \( \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} = \frac{e^{2xy}}{2x} \) confirms the theorem, showcasing the predictability of these derivatives.

Second Order Derivatives

Second order derivatives give us insight into how a function is changing beyond its immediate slope. They indicate the concavity and inflection points of a function.

• For functions of multiple variables, second order derivatives constitute second partial derivatives.
• These derivatives consist of computing the derivative of a derivative, hence you track how rates of change are themselves changing.

In the problem at hand, you found the second order partial derivatives such as \( \frac{\partial^2 z}{\partial x^2} \), \( \frac{\partial^2 z}{\partial y^2} \), and their respective mixed versions. This demonstrates not only the immediate changes in \( z \) with respect to various variables but also showcases the complex interactions, especially visible through the mixed partials.