## Task Condition

Find the derivative.

$$
y=\sqrt{49 x^{2}+1} \cdot \operatorname{arctg} 7 x-\ln \left(7 x+\sqrt{49 x^{2}+1}\right)
$$
## Solution

$y^{\prime}=\left(\sqrt{49 x^{2}+1} \cdot \operatorname{arctg} 7 x-\ln \left(7 x+\sqrt{49 x^{2}+1}\right)\right)^{\prime}=$

$=\frac{1}{2 \sqrt{49 x^{2}+1}} \cdot 7 \cdot \operatorname{arctg} 7 x+\sqrt{49 x^{2}+1} \cdot \frac{1}{1+49 x^{2}} \cdot 7-$

$-\frac{1}{7 x+\sqrt{49 x^{2}+1}} \cdot\left(7+\frac{1}{2 \sqrt{49 x^{2}+1}} \cdot 98 x\right)=$

$=\frac{7 \operatorname{arctg} 7 x}{2 \sqrt{49 x^{2}+1}}+\frac{7}{\sqrt{49 x^{2}+1}}-\frac{1}{7 x+\sqrt{49 x^{2}+1}} \cdot \frac{7 \sqrt{49 x^{2}+1}+49 x}{\sqrt{49 x^{2}+1}}=$

$=\frac{7 \operatorname{arctg} 7 x+14}{2 \sqrt{49 x^{2}+1}}-\frac{7}{\sqrt{49 x^{2}+1}}=\frac{7 \operatorname{arctg} 7 x}{2 \sqrt{49 x^{2}+1}}$

## Problem Kuznetsov Differentiation 13-29