Condition of the problem

Find the $n$-th order derivative.

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

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

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

$\cdots$

Obviously,

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

## Problem Kuznetsov Differentiation 18-6