Condition of the problem

Find the derivative.

$y=(\ln x)^{3^{x}}$
## Solution

$y=(\ln x)^{3^{x}}$

$\ln y=3^{x} \cdot \ln (\ln x)$

$\frac{y^{\prime}}{y}=\left(3^{x} \cdot \ln (\ln x)\right)^{\prime}=3^{x} \cdot \ln 3 \cdot \ln (\ln x)+3^{x} \cdot \frac{1}{\ln x} \cdot \frac{1}{x}=$

$=3^{x} \cdot\left(\ln 3 \cdot \ln (\ln x)+\frac{1}{x \cdot \ln x}\right)$

$y^{\prime}=y \cdot 3^{x} \cdot\left(\ln 3 \cdot \ln (\ln x)+\frac{1}{x \cdot \ln x}\right)=(\ln x)^{3^{x}} \cdot 3^{x} \cdot\left(\ln 3 \cdot \ln (\ln x)+\frac{1}{x \cdot \ln x}\right)$

## Problem Kuznetsov Differentiation 12-5