Problem 17 - Calculus with Concepts in Calculus

Question

Find the first partial derivatives of the function. $$ w=e^{x}(\cos y+\sin z) $$

Step-by-Step Solution

Answer

\( \frac{\partial w}{\partial x} = e^x (\cos y + \sin z) \), \( \frac{\partial w}{\partial y} = -e^x \sin y \), \( \frac{\partial w}{\partial z} = e^x \cos z \).

Step 1: Identify the Function

The function given is \( w = e^x (\cos y + \sin z) \). Identify the function to find its partial derivatives \( \frac{\partial w}{\partial x} \), \( \frac{\partial w}{\partial y} \), and \( \frac{\partial w}{\partial z} \).

Step 2: Find Partial Derivative with Respect to x

To find \( \frac{\partial w}{\partial x} \), treat \( y \) and \( z \) as constants and differentiate with respect to \( x \):\[ \frac{\partial}{\partial x}(e^x (\cos y + \sin z)) = e^x (\cos y + \sin z) \].

Step 3: Find Partial Derivative with Respect to y

To find \( \frac{\partial w}{\partial y} \), treat \( x \) and \( z \) as constants and differentiate with respect to \( y \):\[ \frac{\partial}{\partial y}(e^x (\cos y + \sin z)) = -e^x \sin y \].

Step 4: Find Partial Derivative with Respect to z

To find \( \frac{\partial w}{\partial z} \), treat \( x \) and \( y \) as constants and differentiate with respect to \( z \):\[ \frac{\partial}{\partial z}(e^x (\cos y + \sin z)) = e^x \cos z \].

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing. It is the process of calculating the derivative of a function, which represents the function's slope or gradient. In the context of partial derivatives, which we are dealing with in our exercise, you focus on one variable at a time, while treating the others as constants.
For instance, when we differentiate the given function with respect to \( x \), \( y \), and \( z \), we treat the non-target variables as constants. The idea is the same as ordinary differentiation but extended to functions of several variables. Differentiation is used in various fields such as physics, engineering, and economics to model and predict behavior and changes. It helps in finding tangents to curves and in optimizing real-life applications by identifying maximum and minimum values.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to higher dimensions. It deals with functions of multiple variables. In this exercise, for example, our function \( w = e^x(\cos y + \sin z) \) depends not just on a single variable, but on three: \( x \), \( y \), and \( z \).
Partial derivatives are an essential component of multivariable calculus. They allow us to investigate how changes in one particular variable affect a multivariable function, while keeping others constant. This approach is incredibly useful for analyzing surfaces and optimizing multivariable functions. Understanding how each variable independently contributes is crucial for tackling real-world problems involving multiple interdependent factors.
As with single-variable calculus, multivariable calculus uses tools like limits and differentiation, but the addition of multiple dimensions adds complexity and depth, allowing for more sophisticated models and predictions.

Exponential Functions

Exponential functions are an important class of mathematical functions with the general form \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions exhibit rapid growth and are widely used in various scientific fields, from compound interest in finance to population growth models in biology.
In the given function \( w = e^x(\cos y + \sin z) \), the term \( e^x \) represents the exponential part of the expression. It brings a unique property: when differentiated, the exponential function remains unchanged. This property simplifies differentiation significantly, even when wrapped in more complex expressions as seen in the function we're analyzing.
The stability of exponential functions under differentiation makes them a favorite choice in mathematical modeling. Understanding the behavior of \( e^x \) and how it interacts with other functions, like trigonometric components, is crucial in solving real-world problems effectively.