Problem 105 - Calculus and its applications

Question

A student made the following error on a test: \(\frac{d}{d x} e^{x}=x e^{x-1}\) Identify the error and explain how to correct it.

Step-by-Step Solution

Answer

The derivative of \( e^x \) is \( e^x \), not \( x e^{x-1} \).

Step 1: Identify the Misunderstanding

The student tried to apply the power rule for differentiation, which is incorrect in this context. The function is an exponential function, not a power function. The mistake lies in treating the exponent in a similar way to the power rule: \ \( \frac{d}{dx} x^n = n x^{n-1} \).

Step 2: Recall the Correct Rule for Exponential Functions

To differentiate the exponential function \( e^x \), we need to use the rule for the derivative of exponential functions. The derivative of \( e^x \) with respect to \( x \) is simply itself: \ \( \frac{d}{dx} e^x = e^x \).

Step 3: Apply the Correct Rule

Substitute the correct rule into the expression: \ \( \frac{d}{dx} e^x = e^x \). Hence, the differentiation of the function \( e^x \) remains \( e^x \), and there is no need to multiply by \( x \) or modify the exponent.

Key Concepts

Exponential Functions

Exponential functions are a special class of mathematical functions where the variable is an exponent. The most well-known exponential function is the natural exponential function, expressed as \( e^x \). Here, \( e \) is the base of the natural logarithm and approximately equals 2.71828. Key characteristics of exponential functions include:

• They grow rapidly compared to linear and polynomial functions because as \( x \) increases, \( e^x \) grows faster.
• The rate of growth or decay in exponential functions can be consistently described using its derivative.

Understanding these features is crucial because exponential functions frequently model real-world phenomena such as population growth, radioactive decay, and interest calculations. Each exponential function possesses unique behavior and applications, so it's essential to differentiate them correctly when working with calculus.

Differentiation Rules

Differentiating functions is a fundamental concept in calculus that involves finding a function's derivative, which represents the rate of change. Different rules apply to different types of functions, which is where many students often encounter confusion. For exponential functions like \( e^x \), differentiation is straightforward:

• The derivative of \( e^x \) with respect to \( x \) is simply \( e^x \) itself. This means the rate of change of \( e^x \) is equal to its current value.
• This is a distinct rule and should not be confused with the power rule for polynomials, \( \frac{d}{dx} x^n = nx^{n-1} \).

It is essential to choose the correct differentiation rule depending on the type of function you are dealing with. Correct application ensures solutions are accurate and concepts are well understood.

Common Calculus Errors

Students frequently make mistakes in calculus by applying incorrect differentiation rules. A common error occurs when mixing up rules for differentiating exponential functions and polynomial functions. To avoid these pitfalls:

• Recognize the type of function before differentiating: identify whether it's an exponential function or a polynomial.
• Refresh your knowledge of fundamental differentiation rules regularly to distinguish between them quickly in practice.
• Rewrite the function, if necessary, to clarify which rule should be applied. This practice helps pinpoint possible errors in your approach.

By understanding the specifics of different function types and their rules, you can significantly reduce mistakes, enhancing both your calculus skills and confidence in the subject.