Let $f ( x ) = x ^ { \frac { 1 } { x } }$ . Find the value of $f ^ { \prime } ( e )$ .
A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question

Quizplus · Accepted Answer

The derivative of $f(x) = x^{\frac{1}{x}}$ can be found using logarithmic differentiation. Let $y = x^{\frac{1}{x}}$, then taking the natural logarithm of both sides gives $\ln(y) = \frac{1}{x} \ln(x)$. Differentiating both sides with respect to $x$ gives $\frac{1}{y} \cdot y' = -\frac{1}{x^2} \ln(x) + \frac{1}{x^2}$. Substituting $y = x^{\frac{1}{x}}$ back in and solving for $y'$ gives $y' = x^{\frac{1}{x}} \left(-\frac{\ln(x)}{x^2} + \frac{1}{x^2}\right)$. Substituting $x = e$ gives $y' = e^{\frac{1}{e}} \left(-\frac{1}{e^2} + \frac{1}{e^2}\right) = 0$.

Let f(x)=x1xf ( x ) = x ^ { \frac { 1 } { x } }f(x)=xx1​

Let $f ( x ) = x ^ { \frac { 1 } { x } }$