Problem 51 - Thomas Calculus

Question

Suppose that the function \(v\) in the Product Rule has a constant value \(c .\) What does the Product Rule then say? What does this say about the Constant Multiple Rule?

Step-by-Step Solution

Answer

The Product Rule simplifies to the Constant Multiple Rule: \((uc)' = u'c\).

Step 1: Recall the Product Rule

The Product Rule in calculus is used to find the derivative of the product of two functions. If you have two functions \(u(x)\) and \(v(x)\), the derivative of their product \(u(x)v(x)\) is given by \((uv)' = u'v + uv'\).

Step 2: Substitute the Constant Function for v(x)

In this situation, one of the functions in the product, \(v(x)\), is actually a constant \(c\). Therefore, substitute \(v(x)\) with \(c\) which implies \(v' = 0\).

Step 3: Apply the Product Rule with the Constant

Substituting into the Product Rule formula, we find:\[(uc)' = u'c + u\cdot0 = u'c\]The Product Rule simplifies to \(u'c\) when \(v(x) = c\).

Step 4: Discuss the Constant Multiple Rule

The result from Step 3 is the same as the Constant Multiple Rule, which states that if \(c\) is a constant and \(u(x)\) is a function, then the derivative is \((cu)' = cu'\). This indicates that the Product Rule simplifies directly to the Constant Multiple Rule when one function is constant.

Key Concepts

Constant Multiple Rule

When it comes to calculus, the Constant Multiple Rule is an essential principle for differentiation. It simplifies the process of finding derivatives when one of the terms is a constant. According to this rule, if you have a constant \(c\) multiplied by a function \(u(x)\), the derivative of the function \(cu(x)\) is simply the constant multiplied by the derivative of \(u(x)\). So, mathematically, it is expressed as:

• \((cu)' = cu'\)

This means you can easily "factor out" the constant when differentiating, focusing solely on the function itself. It's like taking out a fixed piece of the puzzle, allowing you to work with the variable parts.
In the context of the Product Rule outlined in the original exercise, applying the Constant Multiple Rule is straightforward when the second function \(v(x)\) is a constant \(c\). The Product Rule naturally collapses into the Constant Multiple Rule, emphasizing how constants streamline differentiation. Utilizing this rule can make the derivative process significantly faster and less error-prone. Its usefulness becomes evident especially in complex equations where constants appear frequently with other variable terms.

Derivative

A derivative represents how a function changes as its input changes. Generally speaking, it's the measure of sensitivity to change of a quantity, which is itself a function of another. In simpler terms, the derivative tells you the rate at which one quantity changes with respect to another.

• In a graphical sense, it is the slope of the tangent line to the function's graph at a particular point.
• Algebraically, it helps in understanding how a small change in the input of a function will result in a change in the output.

Calculating derivatives is a central task in calculus, and we use rules like the Product Rule and Constant Multiple Rule to do so. In the exercise, the derivative of the product \(uv\) is explored where \(v\) is a constant. This simplifies many calculus problems, highlighting how derivatives are influenced by the components of a function.
This foundational concept allows us to solve real-world problems, such as monitoring the speed of an object or determining how the growth rate of investments will change over time. Understanding derivatives is crucial for deeper insights in physics, engineering, economics, and many other fields.

Calculus

Calculus is the mathematical study of change. It consists of two main branches: differential calculus and integral calculus. Differential calculus is concerned with the concept of a derivative, focusing on understanding the rates of change, while integral calculus deals with finding the total size or value such as areas under curves.

• Calculus is fundamentally about analyzing how things change and accumulate.
• It provides the tools to model and predict behaviors in scientific, engineering, and economic systems.

Within the framework of calculus, derivatives help us make sense of dynamic systems by providing insights into local behavior, like how rapidly a car accelerates or how fast a stock is rising.
In the exercise about the Product Rule and Constant Multiple Rule, we're looking specifically at differentiating product functions. Discovering how calculus organizes these principles into clear, applicable rules showcases its power and utility across various applications.