Why the Exponent Must Change
A common power-rule error is to multiply by the exponent but leave the exponent on the variable unchanged. Another version is to change the exponent in the wrong direction. The calculation may look plausible because the exponent appears in the right place as a multiplier, but the resulting function is not the derivative.
The power rule has two distinct jobs: the original exponent becomes a coefficient, and the exponent on the variable decreases by one. Keeping these jobs separate makes the rule easier to remember and makes the most common error easier to catch.
Read the formula from left to right: the old exponent \(r\) is the multiplier, while the new exponent is \(r-1\). For example, differentiating \(x^5\) gives \(5x^4\), not \(5x^5\) and not \(5x^6\).
Why the Exponent Decreases: A Derivation
The difference quotient explains the decrement directly for positive integer powers. For a differentiable function \(f\), its derivative at \(x\) is the limit of \(\frac{f(x+h)-f(x)}{h}\) as \(h\) approaches zero, when that limit exists. Take \(f(x)=x^n\), where \(n\) is a positive integer. The factorization of a difference of powers gives a sum with exactly \(n\) terms:
In the second line, the numerator has been factored using \(a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})\), with \(a=x+h\) and \(b=x\). Since \(a-b=h\), dividing by \(h\) leaves the displayed sum. As \(h\) approaches zero, each of its \(n\) terms approaches \(x^{n-1}\). Adding those \(n\) equal terms produces \(n x^{n-1}\). Thus the exponent decreases because every term remaining after the difference quotient has one fewer factor of \(x\) than the original power.
This derivation proves the rule for positive integer powers. The power rule also applies to fractional and negative powers on suitable domains, as stated above. In each case, the exponent on the resulting variable power is still the original exponent minus one.
Worked Example: A Positive Integer Power
Worked Example: A Positive Integer Power
Differentiate \(f(x)=x^5\). A common incorrect calculation is \(f'(x)=5x^5\): it uses the original exponent as a multiplier but fails to decrement the power.
Apply the power rule with \(r=5\). The multiplier is \(5\), and the new exponent is \(5-1=4\):
Check the result at \(x=2\) using the difference quotient:
Taking the limit as \(h\) approaches zero gives \(80\). The proposed derivative gives \(5(2)^4=5\cdot16=80\), as required. The incorrect expression \(5x^5\) would give \(5(2)^5=160\), so it fails this check.
Worked Example: A Fractional Exponent
Worked Example: A Fractional Exponent
For \(x>0\), differentiate \(g(x)=x^{3/2}\). The exponent is \(3/2\), so it becomes the multiplier; subtracting one gives the new exponent.
In particular, the exponent becomes \(1/2\), not \(3/2\) and not \(5/2\). Check at \(x=4\) from the difference quotient. Let \(t=\sqrt{4+h}\); as \(h\) approaches zero, \(t\) approaches \(2\). Since \(h=t^2-4=(t-2)(t+2)\), we have:
Taking the limit as \(h\) approaches zero, and therefore \(t\) approaches \(2\), gives \(\frac{4+4+4}{2+2}=\frac{12}{4}=3\). The derivative formula gives \(g'(4)=\frac{3}{2}4^{1/2}=\frac{3}{2}\cdot2=3\), which agrees. Leaving the exponent unchanged would give \(\frac{3}{2}4^{3/2}=12\), not \(3\).
Worked Example: A Negative Exponent
Worked Example: A Negative Exponent
For \(x\ne 0\), differentiate \(q(x)=x^{-2}\). Subtracting one from the exponent means \(-2-1=-3\), not \(-2+1=-1\):
Check the value at \(x=2\) using the difference quotient:
As \(h\) approaches zero, this tends to \(-\frac{4}{4(2)^2}=-\frac{4}{16}=-\frac14\). The proposed derivative gives \(q'(2)=-2(2)^{-3}=-\frac{2}{8}=-\frac14\), so the check agrees. Increasing the exponent instead would produce \(-2x^{-1}\), whose value at \(2\) is \(-1\), not \(-\frac14\).
A Reliable Way to Track the Decrement
When applying the power rule, write the exponent subtraction explicitly before simplifying. This makes it harder to copy the old exponent by mistake or to change its sign incorrectly.
In \(x^r\), record the exponent \(r\), including any fraction or negative sign.
Place \(r\) in front of the variable power.
Write \(r-1\) before simplifying the result.
Confirm that it is exactly one less than the original exponent.
For fractional and negative exponents, performing the subtraction with a visible common denominator or signed integer helps. For instance, \(\frac{3}{2}-1=\frac{3}{2}-\frac{2}{2}=\frac{1}{2}\), while \(-2-1=-3\). In both cases, the new exponent is one unit lower than the original.
Common Mistake: Treating the Multiplier as the New Power
The rule contains both \(r\) and \(r-1\), so it is easy to use \(r\) correctly as a multiplier and then mistakenly reuse it as the exponent. For example, the incorrect calculation \(\frac{d}{dx}x^4=4x^4\) keeps the original power. The correction is:
A different mistake is to increase the exponent, as though the result were \(r x^{r+1}\). That is not the power rule: the required change is subtraction by one. A quick numerical check can expose either error. At \(x=1\), several incorrect powers may coincidentally give the same value, so choose a convenient input such as \(x=2\) when the powers give visibly different results. A check at one input is useful for detecting an error, though it does not by itself prove a derivative formula for every input.
Why This Distinction Matters
The multiplier and the new exponent have different roles. The multiplier records how many equal limiting contributions appear in the positive-integer derivation; the reduced exponent describes the remaining power of the variable. Combining those roles into a vague instruction such as “bring down the exponent” can leave the decrement unstated, which is precisely when this error occurs.
Before accepting a power-rule derivative, compare the exponent before and after: if the original exponent is \(r\), the new one must be \(r-1\). This check applies equally when \(r\) is positive, fractional, or negative, subject to the power function being defined and differentiable at the input.
Check Your Understanding
For each question, focus on whether the exponent in the derivative is exactly one less than the original exponent.
- Differentiate \(x^7\). What exponent should appear on \(x\) in the result?
- A student writes \(\frac{d}{dx}x^{5/2}=\frac{5}{2}x^{5/2}\). Identify the error and correct the derivative for \(x>0\).
- What is the result of subtracting one from the exponent \(-4\)? Use it to differentiate \(x^{-4}\) for \(x\ne0\).
- Why does \(r\) appear both as a multiplier and as part of \(r-1\) in the power rule?
- Why can checking a proposed derivative at a single input reveal an error but not prove the formula for all inputs?
