Problem 28 - Calculus Early Transcendentals

Question

\(25-30\) Find the differential of the function. $$ T=\frac{v}{1+u v w} $$

Step-by-Step Solution

Answer

The differential is \( dT = \left( -\frac{v^2vw}{(1+uvw)^2} \right) du + \left( \frac{1}{1+uvw} - \frac{v^2uw}{(1+uvw)^2} \right) dv - \frac{v^2uv}{(1+uvw)^2} dw \).

Step 1: Understand the Problem

We need to find the differential of the function \( T = \frac{v}{1 + uvw} \). The differential \( dT \) tells us how the function \( T \) changes with small changes in its variables \( u, v, \) and \( w \).

Step 2: Define Variables and Differentiation

Consider \( T \) as a function of \( u, v, \) and \( w \). The total differential \( dT \) involves partial derivatives with respect to each variable. We'll calculate \( \frac{\partial T}{\partial u}, \frac{\partial T}{\partial v}, \) and \( \frac{\partial T}{\partial w} \).

Step 3: Compute Partial Derivatives

Use the quotient rule for \( \frac{v}{1 + uvw} \):- \( \frac{\partial T}{\partial u} = -\frac{v^2vw}{(1+uvw)^2} \)- \( \frac{\partial T}{\partial v} = \frac{1}{1+uvw} - \frac{v^2uw}{(1+uvw)^2} \)- \( \frac{\partial T}{\partial w} = -\frac{v^2uv}{(1+uvw)^2} \).

Step 4: Use the Differential Formula

The differential \( dT \) is given by the formula:\[ dT = \frac{\partial T}{\partial u} du + \frac{\partial T}{\partial v} dv + \frac{\partial T}{\partial w} dw \]Substitute the partial derivatives obtained in Step 3.

Step 5: Substitute Partial Derivatives

Substitute the partial derivatives into the differential formula:\[ dT = \left( -\frac{v^2vw}{(1+uvw)^2} \right) du + \left( \frac{1}{1+uvw} - \frac{v^2uw}{(1+uvw)^2} \right) dv - \frac{v^2uv}{(1+uvw)^2} dw \]

Key Concepts

Partial Derivatives

Partial derivatives allow us to understand how a multi-variable function changes with respect to one of its variables while keeping other variables constant.
Imagine you're navigating a hilly terrain. You want to know how steep the hill becomes as you move north (one direction), without considering changes when moving east or west.

This is effectively what a partial derivative does in calculus.

• To compute a partial derivative of a function like \( T = \frac{v}{1 + uvw} \), select one variable, say \( u \), and differentiate while treating other variables \( v \) and \( w \) as constants.
• Repeat this process for all the variables to get \( \frac{\partial T}{\partial u} \), \( \frac{\partial T}{\partial v} \), and \( \frac{\partial T}{\partial w} \).

Partial derivatives are fundamental in finding the total differential of a function, which indicates how the function value changes as all variables change.

Total Differential

The total differential is a concept that tells us how infinitesimal changes in all variables affect the outcome of a function.
Think of it like a recipe where changing the quantity of any ingredient impacts the final taste.

In terms of mathematics, when we have a function \( T = \frac{v}{1+uvw} \):

• Calculate the total differential by using this formula:\[ dT = \frac{\partial T}{\partial u} du + \frac{\partial T}{\partial v} dv + \frac{\partial T}{\partial w} dw \]
• This helps in determining how much and in what direction the function \( T \) will change due to small changes \( du \), \( dv \), and \( dw \) in the variables \( u \), \( v \), and \( w \).

It's especially useful in engineering and physics for approximating outcomes of real-world processes where inputs slightly vary.

Quotient Rule

The quotient rule is an essential tool in calculus used for differentiating functions that are ratios of two other functions.
Remembering the rule is less about memorizing and more about practice.

For any functions \( f(x) \) and \( g(x) \), the quotient rule is:\[ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
In the context of our function, \( T = \frac{v}{1+uvw} \), use the quotient rule to discover how changes in \( u, v, \) and \( w \) impact \( T \):

• Differentiating the numerator and denominator separately.
• Applying the form to find partial derivatives \( \frac{\partial T}{\partial u} \), \( \frac{\partial T}{\partial v} \), and \( \frac{\partial T}{\partial w} \).

Practicing the quotient rule will vastly improve your ability to quickly and accurately work out differentiation problems involving ratios.

Multivariable Functions

Multivariable functions depend on more than one input variable, unlike single-variable functions which depend solely on one.
This adds a layer of complexity but also opens up a richer landscape for exploring relationships within data.

When working with a multivariable function \( T = \frac{v}{1+uvw} \):

• You can investigate how small variations in \( u, v, \) and \( w \) affect the output.
• Total differentials become tools for predicting and analyzing these dependencies.

Understanding these functions is crucial for multiple fields like economics, physics, and any domain involving multi-factorial data analysis.
If you think of a point in 3D space, each coordinate is akin to a variable in a multivariable function, each contributing to its position and resulting effect.