Power Rule with Negative Exponents
Move a denominator power upstairs by changing its exponent to a negative number.
Multiply by the exponent, then subtract one from it.
Keep every restriction that came from the original denominator.
A negative exponent does not require a new derivative rule. It is a compact way to write a reciprocal: \(x^{-n}=1/x^n\). Once a fraction is rewritten in exponent form, the ordinary power rule applies: \(\frac{d}{dx}x^n=nx^{n-1}\). The two moves students need are multiplying by the old exponent and subtracting one from that exponent.
Core example: \(f(x)=x^{-3}\)
- Multiply by the exponent: the coefficient becomes \(-3\).
- Subtract one: \(-3-1=-4\).
- Write the result: \(f'(x)=-3x^{-4}=-3/x^4\), for \(x\ne0\).
Why the exponent becomes more negative
Subtracting one moves any negative exponent farther left on the number line. For example, \(-2-1=-3\), not \(-1\). That is why the derivative of \(1/x^2\) has \(x^3\) in its denominator. A quick sign and size check helps: \(1/x^2\) decreases as positive \(x\) grows, so its derivative should be negative there. The result \(-2/x^3\) passes that check.
A coefficient and several terms
For \(g(x)=5/x^2-3/x+4\), rewrite first: \(g(x)=5x^{-2}-3x^{-1}+4\). Differentiate term by term:
\[g'(x)=-10x^{-3}+3x^{-2}=-\frac{10}{x^3}+\frac{3}{x^2}.\]
The constant becomes zero. Both the function and derivative retain the restriction \(x\ne0\).
Fractional and negative exponents together
Roots in a denominator lead to negative fractional exponents. Since \(1/\sqrt{x}=x^{-1/2}\), its derivative is \(-\tfrac12x^{-3/2}\), which can be written as \(-1/(2x^{3/2})\). Over the real numbers, the original function requires \(x>0\), so the derivative formula is stated on that same domain. Rewriting is useful, but it does not erase restrictions.
When simplifying first is better
If a quotient contains a single monomial denominator, distribute the denominator and use the power rule before reaching for the quotient rule. For example, \((x^3+2x)/x^2=x+2/x\) when \(x\ne0\). Its derivative is \(1-2/x^2\). The quotient rule gives the same result, but it adds algebra and creates more places for a sign error.
Common mistakes
- Adding one instead of subtracting: the rule always uses \(n-1\), even when \(n\) is negative.
- Losing the negative coefficient: \(d(x^{-1})/dx=-x^{-2}\), so \(d(1/x)/dx=-1/x^2\).
- Dropping a domain restriction: rewriting a rational expression does not make \(x=0\) valid.
- Using the quotient rule unnecessarily: rewriting monomial denominators is usually shorter and clearer.
Practice problems with answers
- Differentiate \(4/x^3\).
Show answer
Rewrite as \(4x^{-3}\). The derivative is \(-12x^{-4}=-12/x^4\), with \(x\ne0\).
- Differentiate \(2/x-7/x^2\).
Show answer
\(-2/x^2+14/x^3\), with \(x\ne0\).
- Differentiate \(1/\sqrt{x}\).
Show answer
\(-1/(2x^{3/2})\), for \(x>0\).
For the two most common base cases, compare the derivative of 1/x and the derivative of √x. Then use the Practice Lab for a fresh mixed set.
Reference standard: Rule statements and notation follow OpenStax Calculus Volume 1, Chapter 3. DerivCalc's explanations and examples are independently written.