## Task Condition

Find the derivative.

$y=19^{x^{19}} \cdot x^{19}$
## Solution

$y=19^{x^{19}} \cdot x^{19}$
$\ln y=\ln \left(19^{x^{19}} \cdot x^{19}\right)=\ln \left(19^{x^{19}}\right)+\ln x^{19}=x^{19} \cdot \ln 19+19 \ln x$

$\frac{y^{\prime}}{y}=\left(x^{19} \cdot \ln 19+19 \ln x\right)^{\prime}=19 x^{18} \cdot \ln 19+19 \cdot \frac{1}{x}=$

$=19\left(x^{18} \cdot \ln 19+\frac{1}{x}\right)$

$y^{\prime}=y \cdot 19\left(x^{18} \cdot \ln 19+\frac{1}{x}\right)=19^{x^{19}} \cdot x^{19} \cdot 19\left(x^{18} \cdot \ln 19+\frac{1}{x}\right)$

## Problem Kuznetsov Differentiation 12-19