Problem 35 - Calculus for Biology and Medicine

Question

Find \(\partial f / 2 x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. \(f(x, y, z)=e^{x+y+z}\)

Step-by-Step Solution

Answer

\( \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = \frac{\partial f}{\partial z} = e^{x+y+z} \).

Step 1: Identify the function and variables

The function given is \( f(x, y, z) = e^{x+y+z} \). We need to find the partial derivatives \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \).

Step 2: Calculate \( \frac{\partial f}{\partial x} \)

To find \( \frac{\partial f}{\partial x} \), we treat \( y \) and \( z \) as constants and differentiate with respect to \( x \).The exponent \( x+y+z \) becomes \( y+z \) when taking the derivative concerning \( x \).The derivative is\[\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(e^{x+y+z}) = e^{x+y+z} \cdot 1 = e^{x+y+z}.\]

Step 3: Calculate \( \frac{\partial f}{\partial y} \)

To find \( \frac{\partial f}{\partial y} \), we treat \( x \) and \( z \) as constants and differentiate with respect to \( y \).The derivative is\[\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(e^{x+y+z}) = e^{x+y+z} \cdot 1 = e^{x+y+z}.\]

Step 4: Calculate \( \frac{\partial f}{\partial z} \)

To find \( \frac{\partial f}{\partial z} \), we treat \( x \) and \( y \) as constants and differentiate with respect to \( z \).The derivative is\[\frac{\partial f}{\partial z} = \frac{\partial}{\partial z}(e^{x+y+z}) = e^{x+y+z} \cdot 1 = e^{x+y+z}.\]

Key Concepts

Multivariable Calculus

Multivariable calculus is a branch of mathematics dealing with functions of several variables. It extends the concepts of calculus from functions of a single variable to functions with multiple variables. The real-world applications are vast, including fields like physics, engineering, and economics.
In problems involving more than one variable, each variable might impact the function differently. As such, multivariable calculus allows us to explore these effects through a series of techniques and tools:

• **Partial Derivatives**: They measure how a function changes as one particular variable changes, keeping the others constant. Partial derivatives are foundational in understanding how each variable individually affects the whole system.
• **Gradients**: This is a vector formed by the partial derivatives, pointing in the direction of steepest ascent of the function.
• **Multiple Integration**: Extending techniques from single-variable integration, this helps in calculating volumes and other multi-dimensional measures.

Understanding how to work with multiple dimensions using these tools is essential for solving complex problems in the real world.

Differentiation

Differentiation is a fundamental concept in calculus, used to find the rate at which a quantity changes. For functions of a single variable, this means determining the derivative.
When dealing with functions involving more than one variable, like in our example, the concept expands to partial differentiation.

• **Partial Differentiation**: This involves taking the derivative of a multivariable function with respect to one variable, keeping others constant. This is crucial in scenarios where systems are influenced by multiple factors, helping isolate the effect of one variable at a time.
• **Chain Rule for Partial Derivatives**: A tool used for differentiating compositions of functions is expanded to partial derivatives, allowing the differentiation of complex, multi-layered functions.

Differentiation, particularly partial differentiation, unveils the nuanced changes in each variable, granting detailed insight into mathematical models and their behavior.

Exponential Function

The exponential function, often denoted as \( e^x \), is a fundamental mathematical function with unique properties that make it pivotal in calculus and other scientific fields.
The defining characteristic of exponential functions, such as \( f(x, y, z) = e^{x+y+z} \), is their rapid growth and the property that the function is equal to its own derivative.

• **Exponential Growth and Decay**: Exponential functions describe natural growth processes, from populations to radioactive decay, as they can represent processes where the rate of change is proportional to the current amount.
• **Unique Derivative Property**: In differentiation problems, the function \( e^x \) is its own derivative, meaning \( \frac{d}{dx}(e^x) = e^x \). This property also applies to partial derivatives, maintaining simplicity in differentiation of functions like \( e^{x+y+z} \).

Understanding exponential functions and their properties enables significant insights into both natural phenomena and advanced mathematical models.