The Power Rule and Negative Exponents: A Common Calculus Mistake

The power rule is a fundamental concept in calculus, allowing us to quickly find the derivative of functions involving powers of x. However, a common pitfall arises when dealing with negative exponents. Let's explore this issue and understand how to avoid it.

Understanding the Power Rule

The power rule states that the derivative of xⁿ is nx^(n-1). For example, the derivative of x³ is 3x².

The Pitfall with Negative Exponents

The problem arises when we have negative exponents. Students often mistakenly apply the power rule directly, forgetting the crucial step of rewriting the expression with a positive exponent.

Example:

Let's say we want to find the derivative of x^-2. A common mistake is to directly apply the power rule, resulting in: -2x^-3. However, this is incorrect.

The Correct Approach

The correct approach involves rewriting the expression with a positive exponent before applying the power rule. Remember that x^-n is the same as 1/xⁿ.

Example (continued):

To find the derivative of x^-2, we first rewrite it as 1/x². Now we can apply the power rule:

1. The derivative of 1/x² is -2/x³. (Remember the chain rule since we have a function within a function).

2. This can be rewritten as -2x^-3.

Key Takeaways

Always remember to rewrite expressions with negative exponents as expressions with positive exponents before applying the power rule. This simple step ensures you avoid common errors and achieve accurate results in your calculus calculations.

Further Exploration

For more practice with derivatives and the power rule, check out this YouTube playlist on calculus derivatives: Calculus Derivatives Playlist

Happy learning!