## Task Condition

Find the $n$-th order derivative.

$y=7^{5 x}$
## Solution

$y^{\prime}=5 \ln (7) \cdot 7^{5 x}$

$y^{\prime \prime}=5^{2} \ln (7)^{2} \cdot 7^{5 x}$

$y^{\prime \prime \prime}=5^{3} \ln (7)^{3} \cdot 7^{5 x}$

Assume that $y^{(n)}=5^{n} \ln (7)^{n} \cdot 7^{5 x}$.

We will prove this by mathematical induction.

1. 

$$
n=1 \Rightarrow y^{(n)}=5 \ln (7) 7^{5 x}
$$

2. 
3. 

$$
\begin{aligned}
& n=k-1 \Rightarrow y^{(n)}=5^{(k-1)} \cdot \ln (7)^{(k-1)} \cdot 7^{5 x} \\
& n=k \Rightarrow y^{(n)}=\left(y^{(k-1)}\right)^{\prime}=\left(5^{(k-1)} \ln (7)^{(k-1)} \cdot 7^{5 x}\right)^{\prime}=5^{(k-1)} \ln (7)^{(k-1)} \cdot\left(7^{5 x}\right)^{\prime}=
\end{aligned}
$$

$=5^{(k-1)} \ln (7)^{(k-1)} \cdot\left(5 \ln (7) \cdot 7^{5 x}\right)=5^{k} \ln (7)^{k} \cdot 7^{5 x}$

By the method of mathematical induction, it is proven that $\forall n$ it holds that $y^{(n)}=5^{n} \ln (7)^{n} \cdot 7^{5 x}$

## Problem Kuznetsov Differentiation 18-16