Question

4.9 multiple choice questions
1: let ( f(x)=x^{ln x} ). evaluate ( f^{prime}(e) ). in other words, find the derivative of ( f ) at ( x = e ).
(a) 2 (b) 4 (c) ( e ) (d) ( e^{2} ) (e) none

Explanation:

Step1: Take natural logarithm on both sides

Let \(y = x^{\ln x}\). Then \(\ln y=\ln(x^{\ln x})\). Using the property \(\ln(a^b)=b\ln a\), we get \(\ln y = (\ln x)\cdot(\ln x)=(\ln x)^2\).

Step2: Differentiate both sides with respect to \(x\)

Differentiate \(\ln y = (\ln x)^2\). Using the chain - rule, \(\frac{1}{y}\cdot y'=2\ln x\cdot\frac{1}{x}\). So \(y'=y\cdot\frac{2\ln x}{x}\).
Since \(y = x^{\ln x}\), then \(y'=x^{\ln x}\cdot\frac{2\ln x}{x}\).

Step3: Substitute \(x = e\)

When \(x = e\), \(\ln x = 1\) and \(x^{\ln x}=e^{\ln e}=e^1 = e\).
Substitute into \(y'\): \(y'|_{x = e}=e\cdot\frac{2\times1}{e}=2\).

Answer:

A. \(2\)