Problem 79 - Brief Applied Calculus

Question

Let \(f\) and \(g\) be differentiable functions of \(x\). Assume that denominators are not zero. True or False: \(\frac{d}{d x}(f \cdot g)=f^{\prime} \cdot g^{\prime}\)

Step-by-Step Solution

Answer

False

Step 1: Recall the Product Rule for Derivatives

The product rule states that if you have a differentiable function which is a product of two functions, say \(f(x)\) and \(g(x)\), the derivative of their product is given by \(\frac{d}{dx} (f \cdot g) = f' \cdot g + f \cdot g'\).

Step 2: Compare Given Expression with Product Rule

The expression provided in the exercise is \(\frac{d}{d x}(f \cdot g) = f^{\prime} \cdot g^{\prime}\). According to the product rule, it should instead include the terms \(f' \cdot g + f \cdot g'\), not just \(f' \cdot g'\).

Step 3: Evaluate the Truth of the Statement

Since the expression \(\frac{d}{d x}(f \cdot g) = f^{\prime} \cdot g^{\prime}\) does not match the result from the product rule, which is \(f' \cdot g + f \cdot g'\), the statement provided in the exercise is False.

Key Concepts

Differentiable Functions

A function is considered differentiable if it has a derivative at every point in its domain. This means that you can find the slope of the tangent to the curve of the function at any given point. Essentially, differentiability implies a certain smoothness in the graph of the function. If a function is not differentiable, it might have sharp corners, cusps, or vertical tangent lines.
In mathematics, to say a function like \(f(x)\) is differentiable, it means there exists a derivative, denoted as \(f'(x)\). When working with differentiable functions, it is crucial to remember they satisfy the conditions needed for applying rules of differentiation such as the product rule, quotient rule, and chain rule.
Ensuring both functions \(f(x)\) and \(g(x)\) are differentiable is important for accurately using derivative rules. It allows us to calculate the derivative of their product or quotient conventionally, as no division by zero or undefined behavior occurs.

Calculus

Calculus is the branch of mathematics that studies change. It provides the tools to analyze the rate at which changes occur, often seen in the form of derivatives and integrals. Calculus has two main parts: differential calculus and integral calculus.

• Differential Calculus: Focuses on the concept of the derivative, which represents the rate of change of a quantity.
• Integral Calculus: Deals with the accumulation of quantities and the areas under and between curves.

Understanding calculus concepts opens up insights into understanding motion, growth, and various phenomena in physics, engineering, economics, and beyond.
Within calculus, tools like the product rule, quotient rule, and chain rule help in finding derivatives of more complex functions. These rules are vital for tackling functions composed of simpler elements, making them indispensable for anyone working with calculus.

Derivative

A derivative represents a function's rate of change with respect to a variable. At its core, the derivative answers the question, "how does one quantity change as another changes?" For a function \(f(x)\), the derivative is often denoted as \(f'(x)\) or \(\frac{df}{dx}\).
Derivatives are a fundamental aspect of calculus, capturing how functions behave locally, allowing for the analysis of critical points where maxima or minima occur. They help in the study of a function's increasing or decreasing behavior, concavity, and inflection points.
To compute the derivative of functions that are products, sum, or compositions of simpler functions, calculus provides us with specific rules like the product rule used in the exercise above. According to the product rule, the derivative of the product of two differentiable functions \(f(x)\) and \(g(x)\) is \(f' \cdot g + f \cdot g'\). This fundamental rule ensures that we compute derivatives accurately by accounting for both functions' rate of changes combined.