## Problem Statement

Find the derivative.

$$
y=\frac{2}{3 x-2} \sqrt{-3+12 x-9 x^{2}}+\ln \frac{1+\sqrt{-3+12 x-9 x^{2}}}{3 x-2}
$$
## Solution

Let $z(x)=\sqrt{-3+12 x-9 x^{2}}$. We obtain:

$$
\begin{aligned}
& z'=\frac{1}{2 \sqrt{-3+12 x-9 x^{2}}} \cdot\left(-3+12 x-9 x^{2}\right)'= \\
& =\frac{12-18 x}{2 \sqrt{-3+12 x-9 x^{2}}}=\frac{-3(3 x-2)}{\sqrt{-3+12 x-9 x^{2}}}=\frac{-3(3 x-2)}{z}
\end{aligned}
$$

Then:

$$
\begin{aligned}
& y'=\left(\frac{2}{3 x-2} \sqrt{-3+12 x-9 x^{2}}+\ln \frac{1+\sqrt{-3+12 x-9 x^{2}}}{3 x-2}\right)'= \\
& =\left(\frac{2 z}{3 x-2}+\ln \frac{1+z}{3 x-2}\right)'=\left(\frac{2 z}{3 x-2}\right)'+\left(\ln \frac{1+z}{3 x-2}\right)'= \\
& =2 \frac{z' \cdot(3 x-2)-z \cdot(3 x-2)'}{(3 x-2)^{2}}+\frac{3 x-2}{1+z} \cdot\left(\frac{1+z}{3 x-2}\right)'= \\
& =2 \frac{z' \cdot(3 x-2)-3 z}{(3 x-2)^{2}}+\frac{3 x-2}{1+z} \cdot \frac{(1+z)' \cdot(3 x-2)-(1+z) \cdot(3 x-2)'}{(3 x-2)^{2}}= \\
& =2 \frac{z' \cdot(3 x-2)-3 z}{(3 x-2)^{2}}+\frac{3 x-2}{1+z} \cdot \frac{z' \cdot(3 x-2)-3(1+z)}{(3 x-2)^{2}}= \\
& =2 \frac{\frac{-3(3 x-2)}{z} \cdot(3 x-2)-3 z}{(3 x-2)^{2}}+\frac{3 x-2}{1+z} \cdot \frac{\frac{-3(3 x-2)}{z} \cdot(3 x-2)-3(1+z)}{(3 x-2)^{2}}= \\
& =-\frac{6}{z}-\frac{3 z}{(3 x-2)^{2}}-3 \cdot \frac{3 x-2}{z(1+z)}-\frac{3}{3 x-2}= \\
& =-\frac{6}{\sqrt{-3+12 x-9 x^{2}}}-\frac{3 \sqrt{-3+12 x-9 x^{2}}}{(3 x-2)^{2}}-3 \cdot \frac{3 x-2}{\sqrt{-3+12 x-9 x^{2}}\left(1+\sqrt{-3+12 x-9 x^{2}}\right)}-\frac{3}{3 x-2}= \\
& =-3\left(\frac{2(3 x-2)^{2}}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}+\frac{-3+12 x-9 x^{2}}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}+\right. \\
& \left.+\frac{(3 x-2)\left(1-\sqrt{-3+12 x-9 x^{2}}\right)}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}+\frac{\sqrt{-3+12 x-9 x^{2}}(3 x-2)}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}\right)=
\end{aligned}
$$

$$
\begin{aligned}
& =-3\left(\frac{18 x^{2}-24 x+8-3+12 x-9 x^{2}}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}+\frac{3 x-2}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}\right)= \\
& =-3 \cdot \frac{9 x^{2}-12 x+2}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}=-3 \cdot \frac{(3 x-1) \cdot(3 x-2)}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)^{2}}= \\
& =-3 \cdot \frac{3 x-1}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)}=\frac{3-9 x}{\sqrt{-3+12 x-9 x^{2}}(3 x-2)}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 13-25