If $f(x)=\frac{x}{\ln x}$ , find $f^{\prime}\left(e^{4}\right)$ .
A) $\frac{3}{16}$
B) $\frac{e^{4}}{9}$
C) $\frac{16}{3}$
D) $\frac{4}{3}$
E) $0.4$

Name: If $f(x)=\frac{x}{\ln x}$ , find $f^{\prime}\left(e^{4}\right)$ . A) $\frac{3}{16}$ B) $\frac{e^{4}}{9}$ C) $\frac{16}{3}$ D) $\frac{4}{3}$ E) $0.4$
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Quizplus · Accepted Answer

Using the quotient rule, we have:$$f'(x) = \frac{(\ln x)(1) - x(1/x)}{(\ln x)^2} = \frac{\ln x - 1}{(\ln x)^2}$$So,$$f'(e^4) = \frac{\ln e^4 - 1}{(\ln e^4)^2} = \frac{4 - 1}{4^2} = \frac{3}{16}$$

If f(x)=xln⁡xf(x)=\frac{x}{\ln x}f(x)=lnxx​ , Find f′(e4)f^{\prime}\left(e^{4}\right)f′(e4) A) 316\frac{3}{16}163​ B) e49\frac{e^{4}}{9}9e4​ C) 163\frac{16}{3}316​ D) 43\frac{4}{3}34​ E) 0.40.40.4

If $f(x)=\frac{x}{\ln x}$ , Find $f^{\prime}\left(e^{4}\right)$
A) $\frac{3}{16}$ B) $\frac{e^{4}}{9}$ C) $\frac{16}{3}$ D) $\frac{4}{3}$ E) $0.4$