Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=x / \pi,\) then \(d y / d x=1 / \pi\)

Short answer

The statement is true.

Step-by-step solution

Differentiate the function

The function given is \(y=x/\pi\). Let's differentiate this function with respect to \(x\). Recall that the derivative of \(x\) is \(1\), and a constant coefficient remains the same during differentiation. So, differentiating \(x/\pi\) with respect to \(x\) gives \(\frac{d y}{d x}=\frac{1}{\pi}\)

Compare the result with the statement

Comparing the result from the derivative with the statement \(d y/d x=1/\pi\), it can be seen that they are the same. This means the statement is true.

Key concepts

Derivative of a Function

The derivative of a function is a fundamental concept in calculus, representing the rate at which the function's value changes at any given point. Essentially, it is the instant rate of change or the slope of the tangent line to the function's graph at a specific point. For a function defined by an equation in the form of \(y=f(x)\), the derivative is commonly represented as \(\frac{dy}{dx}\) and is calculated using various differentiation techniques.

For instance, consider the simple linear function in our exercise, \(y=x/\pi\). To find the derivative, or the slope of the line, we identify that \(x\) changes at a constant rate, and the coefficient \(1/\pi\) simply scales this rate. Therefore, the derivative \(\frac{dy}{dx}\) for this function is \(1/\pi\), which indicates that for every unit increase in \(x\), \(y\) increases by \(1/\pi\). This is a direct application of the concept of derivation to determine how quickly the function's value is changing at any point along its graph.

Differentiation Rules

Several rules simplify the process of finding a derivative, known as differentiation rules. Among these are the power rule, product rule, quotient rule, and chain rule. Each of these serves a specific type of function. For example, the power rule is used when differentiating terms like \(x^n\), where \(n\) is any real number.

However, in our exercise, the function is straightforward and doesn't necessarily require these more complex rules. We observe a linear relationship between \(y\) and \(x\), where \(x\) is divided by a constant \(\pi\). In such cases, we rely on simpler differentiation rules, such as the constant coefficient rule, which states that the derivative of a constant times a function is the constant times the derivative of the function. This rule is pivotal to solving our exercise promptly and correctly, as it allows us to treat the constant \(\pi\) separately from the variable \(x\), simplifying the differentiation process.

Constant Coefficient Differentiation

Constant coefficient differentiation is an important rule when dealing with functions that include a constant multiplied by a variable. The rule states that if a function \(y\) can be expressed as a constant \(c\) multiplied by a function of \(x\), \(f(x)\), then its derivative with respect to \(x\) is \(c\) times the derivative of \(f(x)\). Symbolically, this is written as \(\frac{d(c \cdot f(x))}{dx} = c \cdot \frac{df(x)}{dx}\).

In the context of our exercise, the function \(y=x/\pi\) involves the constant coefficient \(1/\pi\). Applying the constant coefficient differentiation rule, we differentiate the variable part, \(x\), which has a derivative of 1, while the constant coefficient \(1/\pi\) remains unchanged. Bringing these two parts together yields \(\frac{1}{\pi}\), thereby confirming the statement's accuracy. Simplification and a clear understanding of this rule can significantly ease the differentiation of more complex functions that involve constants.