Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the first partial derivatives of the function. \(g(u, v)=\frac{u-v}{u+v}\)

Step-by-Step Solution

Answer

The first partial derivatives of the function \(g(u, v) = \frac{u-v}{u+v}\) are: 1. \(\frac{\partial g}{\partial u} = \frac{2v}{(u+v)^{2}}\) 2. \(\frac{\partial g}{\partial v} = \frac{-2u}{(u+v)^{2}}\)

Step 1: Calculate the partial derivative with respect to u

To compute the partial derivative of \(g(u, v)\) with respect to u, we will treat v as a constant and apply the quotient rule. The quotient rule states \(\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}\). Applying the quotient rule, we get: \(\frac{\partial}{\partial u}\) \(\frac{u-v}{u+v} =\frac{((1)(u+v))-((u-v)(1))}{(u+v)^{2}}\)

Step 2: Simplify the expression for (dg/du)

Now we simplify the fraction obtained after applying the quotient rule. \(\frac{\partial g}{\partial u} = \frac{(u + v) - (u - v)}{(u+v)^{2}} = \frac{2v}{(u+v)^{2}}\) After simplifying, the first partial derivative with respect to u is: \( \frac{\partial g}{\partial u} = \frac{2v}{(u+v)^{2}}\) Now, let's compute the partial derivative concerning v.

Step 3: Calculate the partial derivative with respect to v

To compute the partial derivative of \(g(u, v)\) concerning v, we will treat u as a constant and apply the quotient rule. Applying the quotient rule, we get: \(\frac{\partial}{\partial v}\) \(\frac{u-v}{u+v} =\frac{((-1)(u+v))-((u-v)(1))}{(u+v)^{2}}\)

Step 4: Simplify the expression for (dg/dv)

Now we simplify the fraction obtained after applying the quotient rule. \(\frac{\partial g}{\partial v} = \frac{(- u - v) - (u - v)}{(u+v)^{2}} = \frac{-2u}{(u+v)^{2}}\) After simplifying, the first partial derivative with respect to v is: \( \frac{\partial g}{\partial v} = \frac{-2u}{(u+v)^{2}}\) Finally, we have computed the first partial derivatives of the function \(g(u,v)=\frac{u-v}{u+v}\). They are: 1. \( \frac{\partial g}{\partial u} = \frac{2v}{(u+v)^{2}}\) 2. \( \frac{\partial g}{\partial v} = \frac{-2u}{(u+v)^{2}}\)

Key Concepts

Quotient Rule

In calculus, the quotient rule is a method used to find the derivative of a ratio of two functions. It's an essential tool when dealing with functions that are expressed as fractions. The general formula for the quotient rule is given by: \[ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{\left(g(x)\right)^2} \]This formula may look complex at first, but once you break it down, it becomes simpler to understand. Here's how it works:

• Numerator Derivative: Differentiate the numerator function \(f(x)\).
• Denominator Derivative: Differentiate the denominator function \(g(x)\).
• Product and Difference: Multiply the derivative of the numerator by the original denominator and subtract the product of the original numerator and the derivative of the denominator.
• Denominator Squared: Finally, place all of this over the square of the original denominator \(g(x)^2\).

Now when finding partial derivatives like in the original exercise, the quotient rule helps manage the complexity, especially when differentiating with respect to one variable at a time. Remember, while differentiating with respect to one variable, treat other variables as constants.

Function of Two Variables

A function of two variables is a key concept when dealing with multivariable calculus. In such functions, the output depends on two distinct inputs. Let's say we have a function denoted as \(g(u, v)\). The value of \(g\) changes when either \(u\) or \(v\) changes. This is what makes such functions versatile and widely applicable in real-life scenarios, like in physics for describing surfaces, or in economics for multivariable cost functions.Here are some important properties and aspects to consider with functions of two variables:

• Three-Dimensional Graphs: They often represent three-dimensional surfaces in a 3D coordinate space.
• Contours: You can visualize levels of the function as contour maps or level curves on a two-dimensional plane.
• Partial Derivatives: Deriving such functions results in partial derivatives, showing how a function changes with respect to one variable, keeping the other constant.

Understanding this concept is critical when diving into topics related to gradients and optimization, as these functions form the foundation of predictive models and algorithms.

Calculus

Calculus is a branch of mathematics that focuses on rates of change and the accumulation of quantities. It provides tools for modeling systems that change and is widely used across natural sciences, engineering, and economics. Calculus is broadly divided into two areas:

• Differential Calculus: Concerned with the concept of a derivative. It focuses on the rate at which things change – analyzing rates such as speed and growth.
• Integral Calculus: Focuses on accumulation of quantities, such as areas under curves and volumes. It's essentially concerned with the summing up of quantities.

Let's delve further into differential calculus, especially with partial derivatives, as demonstrated in the original exercise:

• Derivatives: They measure sensitivity to change. In single-variable calculus, this involves finding how a function changes as its input changes. In multivariable calculus, this extends to partial derivatives.
• Partial Derivatives: These are used when dealing with functions of several variables. They help analyze the effect of variations in one specific variable, holding others constant.
• Tangent Planes and Linear Approximations: These concepts allow us to create best-fit linear formulas to approximate a multi-variable function around a specific point.

Through understanding both the principles of calculus and techniques like the quotient rule, one can tackle problems involving rates of change with multiple influencing factors.