Mistake Master
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Applying the Power Rule AB & BC

The power rule says $\frac{d}{dx}[x^n] = n x^{n-1}$ for every real $n$, so it covers negative and fractional exponents once the expression is rewritten in that shape: $\sqrt{x}$ as $x^{1/2}$, $\frac{1}{x^3}$ as $x^{-3}$. Subtracting one from a negative exponent makes it more negative, and a constant multiplier rides through unchanged, so $\frac{d}{dx}[7x^3] = 21x^2$.

The rule gets pushed past its boundary in two directions. It is applied to functions with a variable exponent, producing $\frac{d}{dx}[e^x] = xe^{x-1}$ or $\frac{d}{dx}[2^x] = x2^{x-1}$, when the $x$ has to sit in the base for the rule to mean anything. And it is applied to a bare constant, giving $\frac{d}{dx}[5] = 5x^0 = 5$ rather than 0. Alongside these the coefficient gets mishandled, dropped, or multiplied incorrectly into the exponent.

x 5 → 5 x 4 the exponent comes down in front and what is left drops by one n = -3: exponent becomes -4, MORE negative, not less n = 1/2: exponent becomes -1/2, since 1/2 - 1 = -1/2
The same two moves for every real exponent. The only difficulty a negative or fractional n adds is arithmetic.
POWER RULE APPLIES POWER RULE DOES NOT APPLY variable base, constant exponent constant base, variable exponent x⁵, x⁻³, x¹⁄², √x, 1/x⁴ eˣ, 2ˣ, 10ˣ d/dx[xⁿ] = n xⁿ⁻¹ d/dx[eˣ] = eˣ, NOT x eˣ⁻¹ rewrite radicals and reciprocals first their own rules, in Topic 2.7 the test takes one second: find the x. In the base the rule applies, in the exponent not
Same symbols, opposite roles. Nothing about x to the n transfers to n to the x.

The work

3 ways in · any order
Lesson
Applying the Power Rule ›

Applies the power rule across negative and fractional exponents, drills rewriting radicals and reciprocals into power form first, and marks hard the boundary where a variable exponent puts a function outside the rule entirely.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: pushing the power rule onto exponentials or constants where it does not apply, and mishandling the coefficient or the arithmetic of a negative or fractional exponent.

Not yet available · 10 items
Coming soon
Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

Take the diagnostic to identify your misconceptions
Diagnostic first