Question

Exploring a Relationship In Exercises \(71-74\) , the relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime} .\) $$ g(x)=2 f(x) $$

Short answer

If the function \(g(x) = 2f(x)\), its derivative, \(g'(x)\), will be \(2f'(x)\). So, the relationship between \(f'(x)\) and \(g'(x)\) is that the derivative of \(g(x)\) is two times the derivative of \(f(x)\).

Step-by-step solution

Identify the functions

The problem provides us with two functions: \(g(x) = 2f(x)\). \(f(x)\) is the original function and \(g(x)\) is a function that's twice of \(f(x)\). Our goal is to find the relationship between their derivatives.

Understanding the concept of derivatives

The derivative of a function is a concept in calculus that provides us a measure of how a function changes as its input changes. In other words, it's a measure of instantaneous rate of change or the slope of the function at a point.

Find \(f'(x)\) and \(g'(x)\)

We know that if \(g(x) = 2f(x)\), then the derivative of \(g(x)\) with respect to \(x\) is \(g'(x) = 2 * f'(x)\). The *2* is the constant, hence it will remain as it is while differentiating.

Key concepts

Derivatives

At the heart of calculus lies the concept of derivatives, which is a fundamental tool to understand the behavior of functions. Derivatives measure how a function's output changes in response to changes in its input. Essentially, they provide a mathematical snapshot of the function's rate of change at any given point.

For example, if we have a function that describes the position of a car over time, the derivative of this function will tell us the speed of the car at any instant. If we label this position function as \( f(x) \), then we would denote its derivative as \( f'(x) \). In the problem provided, we are working with two functions where one function, \( g(x) \), is simply twice the value of the other, \( f(x) \). Finding \( g'(x) \), the derivative of \( g(x) \), when \( g(x) = 2f(x) \), is a direct application of derivatives. We apply the rules we know, keeping in mind that derivatives distribute over addition and multiplication by a constant factor.

Application to Multiplication by a Constant

When a function is multiplied by a constant, the derivative of that function is also multiplied by the same constant. The process of finding this derivative is straightforward: you simply apply the constant multiple rule. In our example, the derivative of \( g(x) \) would be \( g'(x) = 2f'(x) \), meaning that the rate of change of \( g(x) \) is twice that of \( f(x) \) at any given point. This application is incredibly powerful in the world of calculus, giving us the immediate relationship between \( f'(x) \) and \( g'(x) \) without the need to delve deeper into the functions themselves.

Rate of Change

One might think of derivatives in a more approachable way as measuring the rate of change. Just as a speedometer shows how quickly a car's speed changes, derivatives give us the rate at which function values change with respect to change in input variables. In calculus, this captures the idea of instantaneous rate of change, akin to knowing the exact speed of a car at a precise moment rather than over an extended period of time.

In the textbook exercise, by establishing that \( g(x) = 2f(x) \), we derived that \( g'(x) = 2f'(x) \), which illustrates that the rate at which \( g \) changes is consistently double the rate at which \( f \) changes, no matter the value of \( x \). This is because the rate of change is reflected in the derivative and multiplying the entire function by two scales up its derivative accordingly.

Visualizing Rate of Change

If we were to graph these functions, the slope of the tangent line at any point on \( g(x) \)'s graph would be twice as steep as that on \( f(x) \)'s graph, because the derivative quantifies that slope. This geometric interpretation helps students visualize rates of change and understand how derivatives depict them.

Calculus

Calculus is a branch of mathematics that deals with the calculation of instantaneous rates of change (derivatives) and the accumulation of quantities (integrals). It is a powerful tool that has applications in science, engineering, economics, statistics, and more. Its two main branches are differential calculus and integral calculus, which are interconnected by the fundamental theorem of calculus.

In our exercise, differential calculus is used to examine the relationship between two functions and their derivatives. This process demonstrates how calculus allows us to take general rules and apply them to specific problems to find solutions efficiently. After grasping the derivatives, students will often proceed to learn about integrals, which in many ways, are the reverse of derivatives.

From Concept to Application

A strong understanding of calculus principles enables students to solve complex problems in various disciplines. For instance, in physics, it might be used to describe the motion of particles, while in economics, it could be utilized to optimize production processes. The relationship between \( f(x) \) and \( g(x) \) within our problem is just one example of how calculus simplifies and clarifies the relationships between different mathematical functions.