4.9 multiple choice questions\n1: let ( f(x)=x^{ln x} ). evaluate ( f^{prime}(e) ). in other words, find the…

4.9 multiple choice questions\n1: let ( f(x)=x^{ln x} ). evaluate ( f^{prime}(e) ). in other words, find the derivative of ( f ) at ( x = e ).\n(a) 2 (b) 4 (c) ( e ) (d) ( e^{2} ) (e) none

4.9 multiple choice questions\n1: let ( f(x)=x^{ln x} ). evaluate ( f^{prime}(e) ). in other words, find the derivative of ( f ) at ( x = e ).\n(a) 2 (b) 4 (c) ( e ) (d) ( e^{2} ) (e) none

Answer

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)