Problem 92 - Advanced Calculus

Question

Define the partial derivative of a real valued function, \(\mathrm{f}\). Then compute \((\partial \mathrm{f} / \partial \mathrm{x})\) and \((\partial \mathrm{f} / \partial \mathrm{y})\) where , b) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\cos [\mathrm{x}(1+\mathrm{y})]\), c) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{\mathrm{y}}\) a) \(f(x, y)=x y\)

Step-by-Step Solution

Answer

The partial derivative of a real valued function, f, is the derivative of that function with respect to one of the variables, treating all other variables as constants. For the given functions: a) \( f(x,y) = xy \) \[ \frac{\partial f}{\partial x} = y \quad \text{and} \quad \frac{\partial f}{\partial y} = x \] b) \( f(x, y) = \cos[x(1+y)] \) \[ \frac{\partial f}{\partial x} = -(1+y)\sin[x(1+y)] \quad \text{and} \quad \frac{\partial f}{\partial y} = -x\sin[x(1+y)] \] c) \( f(x, y) = x^y \) \[ \frac{\partial f}{\partial x} = yx^{y-1} \quad \text{and} \quad \frac{\partial f}{\partial y} = (\ln x) x^y \]

Step 1: Define partial derivative

A partial derivative of a function of two or more variables is the derivative of that function with respect to one of the variables, treating all other variables as constants. If the function f(x,y) is assumed to be real valued, the following notation is used for the partial derivatives: 1. Partial derivative with respect to x: \( \frac{\partial f}{\partial x} \) 2. Partial derivative with respect to y: \( \frac{\partial f}{\partial y} \) For given functions a, b, and c, we will now compute their respective partial derivatives. a) \( f(x,y) = xy \)

Step 2: Find \( \frac{\partial f}{\partial x} \) for a

To find the partial derivative with respect to x, we treat y as a constant and differentiate f(x,y) with respect to x: \[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(xy) = y \]

Step 3: Find \( \frac{\partial f}{\partial y} \) for a

To find the partial derivative with respect to y, we treat x as a constant and differentiate f(x,y) with respect to y: \[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(xy) = x \] b) \( f(x, y) = \cos[x(1+y)] \)

Step 4: Find \( \frac{\partial f}{\partial x} \) for b

To find the partial derivative with respect to x, we treat y as a constant and differentiate f(x,y) with respect to x using chain rule: \[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\cos[x(1+y)]) = -(1+y)\sin[x(1+y)] \]

Step 5: Find \( \frac{\partial f}{\partial y} \) for b

To find the partial derivative with respect to y, we treat x as a constant and differentiate f(x,y) with respect to y using chain rule: \[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(\cos[x(1+y)]) = -x\sin[x(1+y)] \] c) \( f(x, y) = x^y \)

Step 6: Find \( \frac{\partial f}{\partial x} \) for c

To find the partial derivative with respect to x, we treat y as a constant and differentiate f(x,y) with respect to x using power and chain rule: \[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^y) = yx^{y-1} \]

Step 7: Find \( \frac{\partial f}{\partial y} \) for c

To find the partial derivative with respect to y, we treat x as a constant and differentiate f(x,y) with respect to y using chain rule and natural logarithm properties: \[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^y) = (\ln x) x^y \] This concludes the computation of the partial derivatives for given functions a, b, and c.

Key Concepts

Chain Rule

The Chain Rule is a foundational concept in calculus, particularly important when dealing with the differentiation of composite functions. It allows us to break the differentiation process into simpler parts and is widely used not only in partial derivatives but also in single-variable functions.
This rule essentially states that if you have a function composed of two functions, say, \( g(x) \) and \( h(x) \), where \( f(x) = g(h(x)) \), then the derivative \( f'(x) \) can be found by multiplying the derivative of \( g \) with respect to \( h \) by the derivative of \( h \) with respect to \( x \).
The formula is:

• \( f'(x) = g'(h(x)) \cdot h'(x) \)

In the context of partial derivatives, the Chain Rule is used when differentiating with respect to one variable while recognizing that the function includes other variables. In Example (b), where \( f(x, y) = \cos[x(1+y)] \), applying the Chain Rule helps us differentiate the inner function, \( x(1+y) \), and then multiply by the derivative of the outer function, \( \cos \). This simplifies complex expressions and makes differentiation manageable.

Differentiation

Differentiation is the action of computing a derivative, which measures how a function changes as its input changes. It's a crucial tool in calculus that helps to understand the behavior of functions in algebraic expressions and graphical representations.
In partial differentiation, we specifically deal with functions of multiple variables. This allows us to determine how a function's value is affected when one of these variables changes, while others remain constant. For instance, in the equation \( f(x, y) = xy \), the derivative \( \frac{\partial f}{\partial x} = y \) shows how \( f \) changes as \( x \) changes alone, treating \( y \) as constant.
To calculate partial derivatives:

• Fix all other variables.
• Apply differentiation rules to the variable of interest.

Partial derivatives are powerful in analyzing time-dependent systems, economic models, and engineering problems involving changes under different conditions.

Real Valued Function

A real-valued function is a function where the output is a real number for every input from its domain. These functions are essential in understanding the mathematical modeling of real-world phenomena because they can represent any quantity that can have real values.
In the context of partial derivatives from our exercises, each function \( f(x, y) \) is real-valued, meaning its output is a real number given real inputs \( x \) and \( y \). Whether we define \( f(x, y) = xy \) or more complex interactions like \( \cos[x(1+y)] \), these are mapped to real numbers.
This gives a concrete interpretation of abstract mathematical concepts, allowing them to be applied in real-world scenarios like physics simulations, economic predictions, or business calculations where variables can represent real quantities such as distance, cost, or time.

Variables in Calculus

Variables in calculus serve as placeholders that represent values or quantities which can either change or remain constant throughout the problem. Understanding how to handle these variables is key to solving calculus problems effectively.
In multivariable calculus, such as in partial derivatives, it's crucial to distinguish between dependent and independent variables. For a function \( f(x, y) \), \( x \) and \( y \) are independent variables which we can alter independently. The dependent variable, typically represented by \( f \), relies on the values that \( x \) and \( y \) take.
When dealing with problems involving these variables:

• Assign what remains constant and what changes.
• Apply methods suited for single-variable or multivariable calculus as necessary.

This handling of variables is foundational in mathematical optimization, physics for force and motion studies, and many other applications where relationships between multiple variables are modeled and analyzed.