Question

Exploring a Relationship In Exercises 79 and \(80,\) verify that \(f^{\prime}(x)=g^{\prime}(x),\) and explain the relationship between \(f\) and \(g .\) $$ f(x)=\frac{3 x}{x+2}, g(x)=\frac{5 x+4}{x+2} $$

Short answer

The derivatives of f(x) and g(x) are equivalent, meaning that their rates of change are the same. The relationship between f(x) and g(x) is simply a vertical transformation, with g(x) being a shift of f(x) upwards by 4 units.

Step-by-step solution

Find the derivative of f(x)

The quotient rule is given by \( (f(x)/g(x))'= \((f'(x)g(x) - f(x)g'(x))/g(x)^2\). Applying this rule to function f(x), \(f'(x)=\((3x)' (x+2) - 3x (x+2)'\) / (x+2)^2 = \((3)(x+2)-(3x)(1)\) / (x+2)^2 = \(3x+6-3x\) / (x+2)^2 = \(6\) / (x+2)^2.

Find the derivative of g(x)

Same with the function g(x), applying the quotient rule gives \(g'(x)=\((5x+4)' (x+2) - (5x+4) (x+2)'\) / (x+2)^2 = \(5x+10-5x-4\) / (x+2)^2 = \(6\) / (x+2)^2.

Verify if f'(x) = g'(x)

By comparing the two derivatives, f'(x) = \(6\) / (x+2)^2 and g'(x) = \(6\) / (x+2)^2. Therefore, f'(x) indeed equals g'(x).

Explain the relationship between f(x) and g(x)

The initial formulas show that f(x) and g(x) are linear transformations of each other. The relationship between f(x) and g(x) is that g(x) is a vertical shift of f(x) by a factor of 4 units upwards. For all x > −2, g(x) is consistently 4 units higher than f(x). This vertical shift does not change the derivative of the function.

Key concepts

Quotient Rule

The quotient rule is an essential technique in calculus used to find the derivative of a function that is expressed as the ratio of two differentiable functions. If you have a function that can be written in the form \( \frac{f(x)}{g(x)} \), where both \(f(x)\) and \(g(x)\) are differentiable, the quotient rule states that the derivative of this compound function is:

\[ \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \]

This formula helps you decompose the rate of change of a division of functions into manageable parts. In practice, to use the quotient rule, you first differentiate the numerator function \(f(x)\) and the denominator function \(g(x)\) separately. Then, you apply the formula by subtracting the product of the derivative of the numerator and the original denominator from the product of the original numerator and the derivative of the denominator, and finally, dividing the result by the square of the original denominator function.

Understanding and applying the quotient rule correctly is crucial for solving complex problems in calculus, particularly when you encounter ratios or fractions of functions. This rule, alongside the product and chain rules, forms the trinity of operations for differentiation.

Calculus

Calculus is a branch of mathematics focused on rates of change (differential calculus) and the accumulation of quantities (integral calculus). It provides a framework for modeling and solving problems involving dynamic systems, which are systems that change over time. Differential calculus, as seen in the quotient rule application, involves taking derivatives to determine the instantaneous rate of change of a function at any point.

A deeper look into differential calculus reveals its applications, ranging from motion and optimization to changes in temperature and rates of reaction in chemistry. In the example exercise, calculus is applied to find the derivatives of functions \(f(x)\) and \(g(x)\), which represents how each function changes as \(x\) changes. More importantly, calculus helps uncover underlying relationships between functions, such as the observed equality of derivatives of \(f(x)\) and \(g(x)\) in the solved exercise. This branch of mathematics not only calculates these rates but also captures the essence of how different functions relate to and affect each other when they undergo transformations or when their inputs vary.

Linear Transformations

Linear transformations refer to operations that take a function or a set of functions and alter them in a way that is linear, which means it's predictable and to a certain extent, reversible. When applied to functions, common linear transformations include scaling (multiplying by a constant), shifting up or down (adding or subtracting a constant), and reflecting across an axis.

In the context of the exercise given, we see that \(g(x)\) is the result of a linear transformation applied to \(f(x)\). Specifically, \(g(x)\) is obtained by adding 4 to the numerator of \(f(x)\), which corresponds to a vertical shift of the graph of \(f(x)\) upward by four units. A key property of linear transformations is that they preserve the operations of addition and scalar multiplication. This means that if two functions are linear transformations of each other, as \(f(x)\) and \(g(x)\) are, their derivatives with respect to \(x\) will remain unchanged, despite the vertical shift. As demonstrated in the given solution, the derivatives of both functions resulted in the same expression, hence, confirming that the vertical shift did not affect the rate of change of the function with respect to \(x\).

Understanding how linear transformations affect functions is vital in various fields of science and engineering. They are a foundational concept in linear algebra and are crucial for solving differential equations, optimizing algorithms, and much more.