## Task Condition

Find the derivative.

$y=x^{2^{x}} \cdot 5^{x}$
## Solution

$y=x^{2^{x}} \cdot 5^{x}$

$\ln y=\ln \left(x^{2^{x}} \cdot 5^{x}\right)=\ln \left(x^{2^{x}}\right)+x \ln 5=2^{x} \cdot \ln (x)+x \ln 5$

$\frac{y^{\prime}}{y}=\left(2^{x} \cdot \ln (x)+x \ln 5\right)^{\prime}=2^{x} \cdot \ln 2 \cdot \ln (x)+2^{x} \cdot \frac{1}{x}+\ln 5$

$y^{\prime}=y \cdot\left(2^{x} \cdot \ln 2 \cdot \ln (x)+2^{x} \cdot \frac{1}{x}+\ln 5\right)=x^{2^{x}} \cdot 5^{x} \cdot\left(2^{x} \cdot \ln 2 \cdot \ln (x)+\frac{2^{x}}{x}+\ln 5\right)$

## Problem Kuznetsov Differentiation 12-24