## Problem Statement

Find the $n$-th order derivative.

$y=a^{2 x+3}$
## Solution

$y=a^{2 x+3}=a^{2 x} \cdot a^{3}=\left(e^{\ln a}\right)^{2 x} \cdot a^{3}=e^{2 x \cdot \ln a} \cdot a^{3}$

$y^{\prime}=\left(e^{2 x \cdot \ln a} \cdot a^{3}\right)^{\prime}=e^{2 x \cdot \ln a} \cdot 2 \ln a \cdot a^{3}$

$y^{\prime \prime}=\left(y^{\prime}\right)^{\prime}=\left(e^{2 x \cdot \ln a} \cdot 2 \ln a \cdot a^{3}\right)^{\prime}=e^{2 x \cdot \ln a} \cdot 2^{2} \cdot \ln ^{2} a \cdot a^{3}$

$\cdots$

Obviously,
$y^{(n)}=e^{2 x \cdot \ln a} \cdot 2^{n} \cdot \ln ^{n} a \cdot a^{3}=a^{2 x} \cdot 2^{n} \cdot \ln ^{n} a \cdot a^{3}=$ $=a^{2 x+3} \cdot 2^{n} \cdot \ln ^{n} a$

## Problem Kuznetsov Differentiation 18-21