Rewrite √x as x^(1/2) before differentiating. Applying the power rule: bring down the exponent (1/2) as a coefficient and subtract 1 from the exponent: f'(x) = (1/2)x^(1/2 − 1) = (1/2)x^(−1/2) = 1/(2√x). The critical first step is the rewrite — trying to differentiate √x without converting it to a fractional exponent has no clear path. This rewriting habit is the main skill the power rule requires beyond the rule itself.
Question 2 Multiple Choice
A student attempts to find d/dx[3^x] by applying the power rule, reasoning that the exponent x should be brought down as a coefficient. What is wrong with this approach?
AThe student should apply the chain rule before the power rule
BThe power rule applies only to integer exponents, so x in the exponent requires a different approach
CThe power rule requires the variable to be the base with a constant exponent; here the variable is in the exponent, making 3^x an exponential function with a different derivative rule
DThe coefficient 3 should be moved to the exponent position first
The power rule d/dx[x^n] = n·x^(n−1) requires the variable to be in the *base* with a *constant* exponent. In 3^x, the base is the constant 3 and the variable x is in the exponent — this is an exponential function, which has a completely different derivative: d/dx[a^x] = a^x · ln(a). Confusing x^n (power function) with a^x (exponential function) is one of the most common errors in early calculus. They look superficially similar but are fundamentally different objects.
Question 3 True / False
The power rule d/dx[x^n] = n·x^(n−1) is valid for n = −3.
TTrue
FFalse
Answer: True
The power rule holds for all real exponents n, including negative integers, fractions, and irrational numbers. For n = −3: d/dx[x^(−3)] = −3·x^(−4) = −3/x^4. You can verify this by rewriting x^(−3) = 1/x^3 and using the quotient rule or limit definition — the result is the same. The extension to negative and fractional exponents is exactly what makes the power rule more powerful than it might first appear.
Question 4 True / False
The derivative of f(x) = x^(1/2) is f'(x) = x^(−1/2).
TTrue
FFalse
Answer: False
Applying the power rule correctly: f'(x) = (1/2)·x^(1/2 − 1) = (1/2)·x^(−1/2). The missing coefficient of 1/2 is the error — the exponent must be brought *down as a multiplier* before the exponent is reduced. The result x^(−1/2) without the factor of 1/2 is wrong. Always write the result as (original exponent) × x^(original exponent − 1).
Question 5 Short Answer
Why can't the power rule be applied to d/dx[2^x], and what does the correct derivative look like?
Think about your answer, then reveal below.
Model answer: The power rule applies only when the variable is the base and the exponent is a constant: f(x) = x^n. In 2^x, the base is the constant 2 and the variable x is the exponent. This makes it an exponential function, not a power function. The correct derivative is d/dx[2^x] = 2^x · ln(2), from the general rule d/dx[a^x] = a^x · ln(a).
The fundamental distinction is which quantity varies. In x^n, the base grows with x while the exponent stays fixed — polynomial-type growth. In a^x, the exponent grows with x while the base stays fixed — exponential growth. The ln(a) factor in the exponential derivative arises from the identity a^x = e^(x ln a): differentiating the exponent x·ln(a) with respect to x gives ln(a), and e^(x ln a) = a^x remains as the factor. If a = e, then ln(e) = 1 and d/dx[e^x] = e^x — the one exponential function that is its own derivative.