First slide

Derivatives

Question

$If y = a^{x^{a^{x^{. . . . .}}}} then the value of \frac{d y}{d x} is$

Moderate

A

$\frac{y^{2}}{x (1 - y \log x)}$
B

$\frac{y^{2} \log y}{x (1 - y \log x)}$
C

$\frac{y^{2} \log y}{x (1 - y \log x \log y)}$
D

$\frac{y^{2} \log y}{x (1 + y \log x \log y)}$

Solution

$\begin{array}{l} y = {a^{x}}^{^{a^{x^{a^{x . . . .}} . \dots}}} \Rightarrow y = a^{x^{y}} \\ \log y = x^{y} \log a \\ \log (\log y) = y \log x + \log (\log a) ∣ \end{array}$
Diff w.r. to ‘x’
$\begin{array}{l} \frac{1}{\log y} \cdot \frac{1}{y} \frac{d y}{d x} = \frac{y}{x} + \log x \frac{d y}{d x} + o \\ \frac{d y}{d x} [\frac{1}{y \log y} - \log x] = \frac{y}{x} \\ \frac{d y}{d x} [\frac{1 - y \log y \log x}{y \log y}] = \frac{y}{x} \Rightarrow \frac{d y}{d x} = \frac{y^{2} \log y}{x [1 - y \log y \log x]} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App