First slide

Derivatives

Question

$If y = {(\frac{1}{x})}^{x} then y_{2} (1) =$

Moderate

A

1
B

-1
C

0
D

2

Solution

$\log y = x \log (1 / x) = - x \log x \Rightarrow \frac{1}{y} y_{1} = - [x \cdot \frac{1}{x} + \log x] \Rightarrow y_{1} = - y (1 + \log x)$
$\begin{array}{l} y_{2} = - [y (\frac{1}{x}) + (1 + \log x) \cdot y_{1}] = - [\frac{y}{x} - y (1 + \log x)^{2}] \\ ∴ y (1) = 1, y_{2} (1) = - [1 - 1] = 0 \end{array}$

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