Discussion
\(\frac{d}{dx}(x^{x})=?\)
Answer: B
The derivative is \(x^{x}(1 + \ln x)\).
Neither the power rule nor the exponential rule applies, because both the base and the exponent vary. Take logarithms first:
\[ \begin{aligned} y &= x^{x} \\ \ln y &= x\ln x \\ \frac{1}{y}\frac{dy}{dx} &= \ln x + 1 \\ \frac{dy}{dx} &= x^{x}(1 + \ln x) \end{aligned} \]
xx-1 wrongly treats the exponent as a constant (power rule), x ln x is ln y itself, and ln x is only part of the derivative of x ln x.
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