## Task Condition

Find the derivative.

$$
y=\sqrt{x^{2}-8 x+17} \cdot \operatorname{arctg}(x-4)-\ln \left(x-4+\sqrt{x^{2}-8 x+17}\right)
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\sqrt{x^{2}-8 x+17} \cdot \operatorname{arctg}(x-4)-\ln \left(x-4+\sqrt{x^{2}-8 x+17}\right)\right)^{\prime}= \\
& =\frac{1}{2 \sqrt{x^{2}-8 x+17}} \cdot(2 x-8) \cdot \operatorname{arctg}(x-4)+\sqrt{x^{2}-8 x+17} \cdot \frac{1}{1+(x-4)^{2}}- \\
& -\frac{1}{x-4+\sqrt{x^{2}-8 x+17}} \cdot\left(1+\frac{1}{2 \sqrt{x^{2}-8 x+17}} \cdot(2 x-8)\right)= \\
& =\frac{x-4}{\sqrt{x^{2}-8 x+17}} \cdot \operatorname{arctg}(x-4)+\frac{\sqrt{x^{2}-8 x+17}}{x^{4}-8 x+17}- \\
& -\frac{1}{x-4+\sqrt{x^{2}-8 x+17}} \cdot \frac{x-4+\sqrt{x^{2}-8 x+17}}{\sqrt{x^{2}-8 x+17}}= \\
& =\frac{x-4}{\sqrt{x^{2}-8 x+17}} \cdot \operatorname{arctg}(x-4)+\frac{1}{\sqrt{x^{2}-8 x+17}}-\frac{1}{\sqrt{x^{2}-8 x+17}}= \\
& =\frac{x-4}{\sqrt{x^{2}-8 x+17}} \cdot \operatorname{arctg}(x-4)
\end{aligned}
$$

## Problem Kuznetsov Differentiation 13-14