Problem 26 - Advanced Engineering Mathematics

Question

Find the first partial derivatives of the given function. $$ f(x, y)=\frac{x y}{\left(x^{2}-y^{2}\right)^{2}} $$

Step-by-Step Solution

Answer

The partial derivatives are complex; simplify as needed using quotient rule formulas.

Step 1: Identify the Variables

The given function is \( f(x, y) = \frac{xy}{(x^2 - y^2)^2} \). Here, \( x \) and \( y \) are the independent variables of the function.

Step 2: Compute the Partial Derivative with respect to x

To find \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant. Use the quotient rule for differentiation: \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\cdot u' - u\cdot v'}{v^2} \). Identify \( u = xy \) and \( v = (x^2 - y^2)^2 \). First, find \( u' = y \) and \( v' = 2\cdot 2(x^2 - y^2) \cdot 2x = 8x(x^2 - y^2) \). Plug these into the quotient rule:\[\frac{\partial f}{\partial x} = \frac{(x^2 - y^2)^2 \cdot y - xy \cdot 8x(x^2-y^2)}{((x^2-y^2)^2)^2}\]

Step 3: Simplify the Partial Derivative with respect to x

Simplify the expression from Step 2:\[\frac{\partial f}{\partial x} = \frac{y(x^2-y^2)^2 - 8x^2y(x^2 - y^2)}{(x^2-y^2)^4}\]Factor out common terms if possible:\[= \frac{y((x^2-y^2)^2 - 8x^2(x^2-y^2))}{(x^2-y^2)^4}\]

Step 4: Compute the Partial Derivative with respect to y

To find \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant. Again, apply the quotient rule with \( u = xy \) and \( v = (x^2 - y^2)^2 \). Here, \( u' = x \) and \( v' = 8y(x^2-y^2) \). Substitute into the quotient rule:\[\frac{\partial f}{\partial y} = \frac{(x^2-y^2)^2 \cdot x - xy \cdot 8y(x^2-y^2)}{(x^2-y^2)^4}\]

Step 5: Simplify the Partial Derivative with respect to y

Simplify the expression from Step 4:\[\frac{\partial f}{\partial y} = \frac{x(x^2-y^2)^2 - 8xy^2(x^2-y^2)}{(x^2-y^2)^4}\]Factor out any common terms if possible:\[= \frac{x((x^2-y^2)^2 - 8y^2(x^2-y^2))}{(x^2-y^2)^4}\]

Key Concepts

Quotient Rule

The quotient rule is a powerful differentiation technique used when you have a function that is the ratio of two other functions. This is especially important in situations involving partial derivatives where each variable can affect the numerator and denominator. The rule is expressed as:

• For functions of the form \( \frac{u}{v} \), the derivative \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \).

In the context of partial derivatives, like in the original exercise, understanding which variables to treat as constants is crucial. When computing \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant, and vice versa for \( \frac{\partial f}{\partial y} \). This differentiation method helps simplify complex fractions by systematically considering the rate of change of both \( u \) and \( v \). Remember to simplify your final result by canceling out common terms, reducing to an easier-to-understand form.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of more than one variable. This is essential for analyzing systems where multiple influences can affect outcomes, such as in physics and engineering. In our function \( f(x, y) = \frac{xy}{(x^2 - y^2)^2} \), we deal with two independent variables: \( x \) and \( y \). This means the function's behavior depends on both these variables simultaneously.

• Applications include optimizing processes involving multiple parameters.
• It is crucial even for drawing graphs of functions that exist in a multidimensional space.

Using partial derivatives, we find how the function changes when one variable changes, holding others constant. This approach is fundamental in fields like economics for utility functions or in meteorology for climate models. Understanding these changes helps in comprehending the function's sensitivity and interaction dynamics at various points.

Differentiation Techniques

Differentiation techniques are methods used in calculus to find the derivative of a function, which represents the function's rate of change. For multivariable functions, these techniques extend beyond the basic rules used in single-variable functions. Here are some key techniques:

• Partial Differentiation: Focus on one variable while treating others as constants.
• Quotient Rule: Valuable for functions that present as a fraction, as seen in our example.
• Chain Rule: Useful when dealing with composition of functions, though not used specifically in this exercise.

In partial differentiation, applying the quotient rule requires careful attention to each component's derivative with respect to the variable of interest. This careful approach ensures accurate calculations, vital for precise application in real-world problem-solving scenarios. Always check your work by simplifying expressions and identifying any potential mistakes early in the process.