Mathematics
The Power Rule and Its Extension to Negative Exponents
Quick fact
The power rule works for any real exponent, not just positive integers. So d/dx(x^π) is simply π·x^(π−1), just as easily as for x².
Why this is interesting
You've mastered differentiating x², but what about 1/x²? The same simple rule works—surprisingly—even when exponents are negative. How is that possible?
Read the full explanation
Understanding The Power Rule and Its Extension to Negative Exponents
Let's start with what you already know: the derivative of x² is 2x. The pattern? Multiply by the exponent (2) and reduce it by one (2 → 1). This is the power rule: for any positive integer n, the derivative of x^n is n·x^(n−1). But what if the exponent is negative, like x⁻²? That's just 1/x². You might think you need a different rule, but the power rule applies to negative exponents just as well. That means the derivative of x⁻² is -2·x⁻³. You can verify this by differentiating 1/x² using the quotient rule or by rewriting it as x⁻² and using the power rule—you'll get the same result.
A deeper explanation
The power rule emerges from the definition of the derivative as a limit. For positive integer exponents, you can expand (x+h)^n using the binomial theorem; the h term gives a coefficient n·x^(n−1), which becomes the derivative as h→0. For negative exponents, the rule can be proven using the quotient rule or by extending the algebra to negative integers through the laws of exponents. The key insight is that the derivative of x^n is n·x^(n−1) for any real n, which is a theorem proved rigorously using limits. This extension is crucial because it allows you to differentiate reciprocal functions without memorizing a separate rule, simplifying calculations in physics (e.g., inverse-square laws) and economics (e.g., marginal rates).