Question

Exploring a Relationship In Exercises \(71-74\) , the relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime} .\) $$ g(x)=-5 f(x) $$

Short answer

The relationship between the derivatives of \(f\) and \(g\) is \(g'(x)=-5f'(x)\)

Step-by-step solution

Relationship of f and g in terms of their derivatives

The relationship between the two functions we have is given by \(g(x)=-5 f(x)\). To find the relationship between their derivatives, we differentiate both sides with respect to \(x\). Remember the rule of differentiation that says the derivative of a constant times a function, is the constant times the derivative of the function.

Differentiate each Function

The differentiation of \(g(x)=-5 f(x)\) with respect to \(x\) gives us \(g'(x)=-5f'(x)\). This implies that the derivative of function \(g(x)\) is equal to -5 times the derivative of function \(f(x)\).

Key concepts

Relationship Between Derivatives

Understanding the relationship between derivatives is a fundamental aspect of differential calculus. When we are given two functions that are related, like in the equation g(x) = -5 f(x), their derivatives will also exhibit a specific relationship. To see the connection, we perform differentiation, which reveals how the rate of change of one function is related to the rate of change of another. As demonstrated in the exercise, finding the derivatives of both sides of the given equation with respect to x yields g'(x) = -5f'(x). This outcome illustrates a critical point: the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function. This ties directly into how scalars interact with the notion of 'rate of change' in calculus.

• Proportionality: If one function is a scalar multiple of another, their derivatives maintain that proportionality.
• Inversely Related: When the scalar is negative, the functions increase or decrease in opposite directions.
• Consistent Relationship: The relationship is maintained regardless of the complexity of the original functions.

Differential Calculus

Differential calculus is a branch of mathematics predominantly concerned with rates of change and the slope of curves. When we talk about the 'derivative' of a function, we refer to a measure of how the function's output value changes as its input value changes. This concept is crucial not only in physics for describing motion but in a myriad of fields where understanding dynamic change is important.

For example, when a function f(x) represents the distance traveled over time, its derivative f'(x), is the velocity, describing how quickly the distance changes over time. Differential calculus provides the tools to analyze and predict such behaviors mathematically by dealing with variables and their infinitesimal differences – a core idea in calculus known as the limit.

Chain Rule of Differentiation

The chain rule is an essential theorem in differential calculus used to differentiate composite functions. A composite function can be thought of as a function within another function—something which occurs commonly in mathematical and real-world applications. In essence, the chain rule helps us break down the differentiation process into simpler parts when faced with such compositions.

Imagine you have a function h(x) = f(g(x)), the chain rule states that the derivative of h(x), shown as h'(x), is the product of the derivative of f with respect to g(x) and the derivative of g with respect to x. Symbolically, this is written as h'(x) = f'(g(x)) * g'(x). It allows us to systematically take derivatives of functions that are layered on each other, providing clarity and ease in complex situations.

Calculus of a Single Variable

Calculus of a single variable, often the first calculus encountered by students, deals with functions that have one independent variable. It lays the foundation for all of calculus, introducing the concepts of derivatives and integrals in relation to a sole varying quantity. This branch of calculus is concerned with understanding the behavior of functions graphically and analytically through their rates of change and accumulated quantities.

In the example from the exercise, g(x) and f(x) are functions of a single variable x, and when we differentiate them, we're looking at how they change with respect to that one variable. The calculus of a single variable is foundational for more advanced topics, such as multivariable calculus, which extends these principles to functions of more than one variable.