Problem 37 - Calculus for Biology and Medicine

Question

In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=\ln (x+y+z) $$

Step-by-Step Solution

Answer

The partial derivatives are \( \frac{1}{x+y+z} \) for each \( \partial f / \partial x \), \( \partial f / \partial y \), and \( \partial f / \partial z \).

Step 1: Identifying Partial Derivatives

The function provided is a multivariable function, specifically a function of three variables: \( f(x, y, z) = \ln(x + y + z) \). We need to find the partial derivatives of this function with respect to \( x \), \( y \), and \( z \). The partial derivative with respect to a particular variable is found by treating that variable as a variable and all other variables as constants.

Step 1: Calculating \( \partial f / \partial x \)

To find \( \partial f / \partial x \), differentiate \( f = \ln(x + y + z) \) with respect to \( x \). The derivative of \( \ln(u) \) is \( \frac{1}{u} \) times the derivative of \( u \). Here, \( u = x + y + z \). Thus, \( \frac{d}{dx}[x+y+z] = 1 \). Therefore, \( \partial f / \partial x = \frac{1}{x+y+z} \times 1 = \frac{1}{x+y+z} \).

Step 2: Calculating \( \partial f / \partial y \)

Now, calculate \( \partial f / \partial y \) by differentiating \( f = \ln(x + y + z) \) with respect to \( y \). Using a similar approach as in Step 1, \( u = x + y + z \) and \( \frac{d}{dy}[x + y + z] = 1 \). Hence, \( \partial f / \partial y = \frac{1}{x+y+z} \times 1 = \frac{1}{x+y+z} \).

Step 3: Calculating \( \partial f / \partial z \)

Finally, find \( \partial f / \partial z \) by differentiating \( f = \ln(x + y + z) \) with respect to \( z \). Again, \( u = x + y + z \) and \( \frac{d}{dz}[x + y + z] = 1 \). Therefore, \( \partial f / \partial z = \frac{1}{x+y+z} \times 1 = \frac{1}{x+y+z} \).

Key Concepts

Multivariable Function

In mathematics, a multivariable function is a type of function that involves more than one input or independent variable. Functions like these are common in calculus and occur frequently in scientific fields that model phenomena involving several factors.
For example, the function \( f(x, y, z) = \ln(x + y + z) \) is a function of three variables: \( x \), \( y \), and \( z \). Here's why multivariable functions are important:

• They describe systems with multiple influencing factors, much like real-life scenarios.
• They enable us to model relationships where changes in one variable might affect the outcome differently than changes in another.
• Understanding how each variable influences the output is crucial for optimization and prediction, which is where partial derivatives come into play.

To analyze these functions, we often employ techniques from multivariable calculus, which allows us to grasp how changes in each variable alter the function's overall value.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function's output value changes as its input changes. It helps in understanding how a function behaves and provides a mechanism to find rates of change. In the context of multivariable functions, this concept is extended to partial differentiation.
Partial differentiation is the process of finding the derivative of a multivariable function with respect to one of its variables while treating the other variables as constants. This allows us to understand how a change in one specific variable affects the function’s output, independently of the other variables.

• The partial derivative \( \frac{\partial f}{\partial x} \) tells us the rate at which the function \( f \) changes as \( x \) changes, with \( y \) and \( z \) held constant.
• Similarly, \( \frac{\partial f}{\partial y} \) and \( \frac{\partial f}{\partial z} \) indicate the sensitivity of the function to changes in \( y \) and \( z \), respectively.

This is especially useful when dealing with real-world problems, such as finding the gradient of a surface at a given point, understanding the direction of quickest ascent, or optimizing for the best outcome.

Calculus

Calculus is a branch of mathematics that explores how quantities change and accumulate. It's crucial for understanding motion, area, volume, and so much more in applied mathematics and science. In dealing with multivariable functions, calculus not only involves finding derivatives but also handling integrals.
In the specific case of differentiation, calculus introduces tools like the product rule, chain rule, and quotient rule, which help in dealing with functions beyond simple polynomials. For multivariable functions, the concept of partial differentiation expands these to accommodate multiple dimensions.

• The **Chain Rule** is particularly useful when differentiating composite functions. It helps in expressing the derivative of a composite function in terms of the derivatives of its components.
• Another aspect of calculus is **Integration**, which often comes after differentiation. While we differentiate to find rates, we integrate to find the accumulation or total change. For multivariable functions, this can lead to double or triple integrals.

Mastering calculus allows us to navigate through complex problems involving rates of change and total accumulation in a structured way, providing a foundation for advanced mathematical, physical, and engineering applications.