Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Is there an example of two functions, \(u(x)\) and \(v(x),\) for which \([u(x) \times v(x)]^{\prime}=u^{\prime}(x) \times v^{\prime}(x) ?\)

Step-by-Step Solution

Answer

Yes, for constant functions like \(u(x) = c\) and \(v(x) = c\), this condition is satisfied.

Step 1: Interpreting the Derivative Condition

We are given the condition \([u(x) \times v(x)]^{\prime} = u^{\prime}(x) \times v^{\prime}(x)\). According to the product rule, \([u(x) \times v(x)]^{\prime} = u^{\prime}(x) v(x) + u(x) v^{\prime}(x)\). We need to check if this matches with \(u^{\prime}(x) \times v^{\prime}(x)\).

Step 2: Testing the Zero Function Example

Consider \(u(x) = c\) and \(v(x) = c\), where \(c\) is a constant function. Then \(u^{\prime}(x) = 0\) and \(v^{\prime}(x) = 0\). So, \([u(x) \times v(x)]^{\prime} = (c \times c)^{\prime} = 0\), and \(u^{\prime}(x) \times v^{\prime}(x) = 0 \times 0 = 0\). They are equal, meeting the condition.

Step 3: Verifying Another Possible Case

If both functions are constant, for instance, choose \(u(x) = 1\) and \(v(x) = 1\). Then, \(u^{\prime}(x) = 0\) and \(v^{\prime}(x) = 0\). Hence, \([u(x) \times v(x)]^{\prime} = (1 \times 1)^{\prime} = 0\), and \(u^{\prime}(x) \times v^{\prime}(x) = 0 \times 0 = 0\). Therefore, the condition holds true.

Key Concepts

Product Rule in Calculus

The product rule is an essential tool in calculus used when differentiating two functions that are multiplied together. It helps to find the derivative of a product of functions. Imagine you have two functions, \(u(x)\) and \(v(x)\), and they are combined as \(u(x) \times v(x)\). Whereas computing the derivative of each function separately is straightforward, doing so for their product requires the product rule. It states that:

• \([u(x) \times v(x)]' = u'(x) v(x) + u(x) v'(x)\)

This formula might look complex, but it mirrors a fundamental dynamic: the changing nature of each component contributes to the rate of change of the product. If either function changes, the product itself will change, capturing how these changes interact.
The product rule is particularly valuable because it allows for the differentiation of products without needing to expand the product first. This is especially helpful when you have complex functions, allowing you to sidestep extensive algebraic manipulations.

Understanding Differentiation

Differentiation refers to the process of finding the derivative of a function, which effectively measures how a function changes as its input changes. The derivative is like a rate of change or the slope of the function at any given point. Imagine driving a car: the speed at which you move is the derivative of your position relative to time.
The standard notation for derivatives involves using a prime symbol like \(u'(x)\) or \(v'(x)\). To differentiate a function means to apply specific rules, such as the product rule, chain rule, or quotient rule, depending on the situation.

• The power rule, for example, is for differentiating simple polynomials.
• The product rule is used for multiplying functions.
• The chain rule is applied for composite functions.

For a consistent check on learning, remember that differentiation transforms functions into their derivative form, yielding valuable insights into their behavior. Differentiation helps determine slopes, analyze growth trends, and solve problems involving rates, among other applications in mathematics and science.

Exploring Functions and Their Properties

Functions represent the relationship between two variables, usually denoted as \(x\) (the independent variable) and \(f(x)\) or \(y\) (the dependent variable). Functions can take many forms — they may be linear, quadratic, exponential, or something more complex. Knowing the type and properties of a function, such as continuity or differentiability, is fundamental in calculus.
Functions often have unique characteristics that define their graphs, behavior, and applications. Consider these basics:

• Linear functions have a constant rate of change and form straight lines.
• Quadratic functions create parabolas, impacting various fields like physics with projectile motion.
• Exponential functions rapidly increase or decrease, critical in studies of growth and decay.

Understanding the behavior of functions is pivotal for correctly applying differentiation rules like the product rule. When solving calculus problems involving multiple functions, recognizing how they work individuall and together ensures precise and effective solutions.