## Task Condition

Find the $n$-th order derivative.

$$
y=\frac{4+15 x}{5 x+1}
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\frac{4+15 x}{5 x+1}\right)^{\prime}=\frac{15 \cdot(5 x+1)-(4+15 x) \cdot 5}{(5 x+1)^{2}}=\frac{75 x+15-20-75 x}{(5 x+1)^{2}}= \\
& =-\frac{5}{(5 x+1)^{2}} \\
& y^{\prime \prime}=\left(y^{\prime}\right)^{\prime}=\left(-\frac{5}{(5 x+1)^{2}}\right)^{\prime}=-\frac{5 \cdot(-2)}{(5 x+1)^{3}} \cdot 5=\frac{2 \cdot 5^{2}}{(5 x+1)^{3}} \\
& y^{\prime \prime \prime}=\left(y^{\prime \prime}\right)^{\prime}=\left(\frac{2 \cdot 5^{2}}{(5 x+1)^{3}}\right)^{\prime}=\frac{2 \cdot(-3) \cdot 5^{2}}{(5 x+1)^{4}} \cdot 5=-\frac{2 \cdot 3 \cdot 5^{3}}{(5 x+1)^{4}}
\end{aligned}
$$

Obviously,

$y^{(n)}=\frac{(-1)^{n} \cdot n!\cdot 5^{n}}{(5 x+1)^{n+1}}$

## Problem Kuznetsov Differentiation 18-14