Problem 62 - Applied Calculus

Question

Show that for any power function \(f(x)=x^{n}\), we have \(f^{\prime}(1)=n\)

Step-by-Step Solution

Answer

The derivative \( f'(1) = n \) for \( f(x) = x^n \).

Step 1: Understand the Problem

We need to show that the derivative of the function \( f(x) = x^n \) evaluated at \( x = 1 \) is equal to \( n \). This means finding \( f'(x) \) and then substituting \( x = 1 \) into this derivative.

Step 2: Recall the Power Rule for Derivatives

The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) \) is found using the formula \( f'(x) = nx^{n-1} \). This formula is applicable for any real number \( n \).

Step 3: Apply the Power Rule

Using the power rule, we differentiate the function \( f(x) = x^n \) to get \( f'(x) = nx^{n-1} \).

Step 4: Evaluate the Derivative at x = 1

Substitute \( x = 1 \) into the derivative \( f'(x) = nx^{n-1} \). We have \( f'(1) = n(1)^{n-1} = n \), since \( 1^{n-1} = 1 \) for any \( n \).

Step 5: Conclusion

We have shown that \( f'(1) = n \) for the power function \( f(x) = x^n \), as required by the problem statement.

Key Concepts

Derivatives

To start with the concept of derivatives, it is essential to understand that a derivative of a function essentially tells us how the function's value changes as the input changes. When we talk about derivatives, we are often referring to the slope of the function's graph at a particular point. For a function given by \( f(x) = x^n \), the derivative \( f'(x) \) represents the rate at which \( f(x) \) changes with respect to \( x \).

Derivatives are foundational in calculus, because they give us powerful tools to analyze and interpret functions. By knowing how a function behaves in relation to its input, we are able to make predictions about the function's tendencies, such as increasing or decreasing behaviour. Understanding derivatives is crucial for solving many real-world problems, such as optimization and curve sketching.

In our context, the derivative helps determine the nature of the polynomial \( x^n \) at specific points, such as \( x = 1 \).

Power Rule

The power rule is one of the simplest yet most useful rules for finding derivatives. It applies when we have a power function, which is a function in the form \( f(x) = x^n \). In essence, the power rule tells us how to quickly differentiate such functions.

• If \( f(x) = x^n \), then its derivative, denoted \( f'(x) \), is \( nx^{n-1} \).

This means that to find the derivative, we multiply the power \( n \) by the term \( x \) raised to the one less power, \( n-1 \).

For example, if \( n = 3 \), the derivative of \( x^3 \) will be \( 3x^2 \). This simplification is incredibly handy because it allows you to differentiate power functions rapidly without dealing with limits directly every time.

By using the power rule, it becomes straightforward to find the derivative of any power function and analyze its behavior confidently at any point, such as \( x = 1 \), in our case.

Evaluating Derivatives

Evaluating derivatives means finding the specific value of the derivative at a certain point. This process involves two main steps: first, finding the general derivative of the function, and second, substituting the specific point into this derivative to evaluate it.

In our example with the power function \( f(x) = x^n \), we first used the power rule to find that \( f'(x) = nx^{n-1} \). For evaluation, we substitute \( x = 1 \) into this expression, giving us \( f'(1) = n(1)^{n-1} \).

• Since \( 1^{n-1} = 1 \) for any \( n \), the result simplifies to \( f'(1) = n \).

Therefore, by evaluating the derivative at a particular point, we get the instantaneous rate of change of the function at that specific point. This is particularly useful when solving problems that involve real-world applications where exact rates or slopes at certain points are required.