Problem 10 - Calculus and its applications

Question

Find \(f_{x}\) and \(f_{y}\). $$f(x, y)=e^{2 x-y}$$

Step-by-Step Solution

Answer

\(f_x = 2e^{2x-y}\), \(f_y = -e^{2x-y}\).

Step 1: Identify the Partial Derivative with Respect to x

To find the partial derivative of the function \( f(x, y) = e^{2x - y} \) with respect to \(x\), treat \(y\) as a constant and differentiate the function as you would with a single-variable function of \(x\).

Step 2: Differentiate with Respect to x

The derivative of \(e^{2x - y}\) with respect to \(x\) is obtained by applying the chain rule. The derivative of the exponent \(2x - y\) with respect to \(x\) is \(2\). Therefore, \(f_x = \frac{d}{dx}(e^{2x - y}) = e^{2x - y} \cdot 2 = 2e^{2x - y}\).

Step 3: Identify the Partial Derivative with Respect to y

Now, to find the partial derivative with respect to \(y\), treat \(x\) as a constant and differentiate the function as you would with a single-variable function of \(y\).

Step 4: Differentiate with Respect to y

The derivative of \(e^{2x - y}\) with respect to \(y\) is found by applying the chain rule. The derivative of the exponent \(2x - y\) with respect to \(y\) is \(-1\). So, \(f_y = \frac{d}{dy}(e^{2x - y}) = e^{2x - y} \cdot (-1) = -e^{2x - y}\).

Key Concepts

Understanding Partial Derivatives

Partial derivatives are a fascinating aspect of calculus, especially useful in functions involving multiple variables. When we talk about partial derivatives, we're referring to the derivatives of functions that depend on more than one variable, like functions of the form \( f(x, y) \). The key idea here is that when taking a partial derivative with respect to one variable, the other variables are treated as constants.

Here's how it works in simple terms:

• Identify the variable to differentiate: When finding \( f_x \), consider how only changes in \( x \) affect \( f \), treating \( y \) as a constant.
• Apply derivative rules: Differentiate as if it's a single-variable function. The partial derivative with respect to \( x \) captures the rate of change of the function as only \( x \) changes, holding \( y \) fixed.
• Real-world examples: Partial derivatives are crucial in physics and engineering, measuring things like sensitivity of systems and optimizing multi-variable functions.

Partial derivatives help us understand the behavior of complex functions by breaking them down into more manageable pieces.

The Chain Rule Simplified

The chain rule is a fundamental derivative rule that you often encounter in calculus. It is the method we use to differentiate compositions of functions, that is, functions within functions. This is especially handy when dealing with problems relating to partial derivatives.

The chain rule allows us to easily handle these complex tasks. In our case with the exponential function \( e^{2x-y} \):

• Clarify the setup: Recognize \( e^{2x-y} \) as a composition of the exponential function and its exponent \( 2x - y \).
• Differentiate the inner function: First, differentiate \( 2x - y \) with respect to the chosen variable, like \( x \) or \( y \). For example, for \( x \), the derivative of \( 2x - y \) is \( 2 \).
• Multiply these derivatives: The result is then multiplied by the derivative of the outer function (here it's simply the exponential \( e^{2x-y} \)). Thus, for \( f_x \), you get \( 2e^{2x-y} \).

With the chain rule, tackling nested functions becomes intuitive and straightforward.

Exponential Functions Demystified

Exponential functions are integral in many scientific fields and form the backbone of growth processes, like population growth or radioactive decay. A typical exponential function follows the form \( f(x) = e^{g(x)} \), where \( e \) is the base of the natural logarithm, roughly equal to 2.71828.

These functions are distinct due to their rapid growth. In calculus, differentiating exponential functions is relatively straightforward, allowing them to maintain their original form even after differentiation:

• Characteristics: Exponential functions like \( e^{x} \) are unique because their derivative is the same as themselves.
• Applications: We see them in fields requiring modeling of continuous growth patterns, such as finance, biology, and physics.
• Simplified differentiation: When you differentiate \( e^{u} \), you get \( e^{u} \cdot u' \), where \( u' \) is the derivative of the exponent, thanks to the chain rule.

This makes exponential functions both powerful and versatile for calculus and beyond.