Cho hàm số y=f(x) = căn (tan x+ cot x) . Giá trị bằng

* Hướng dẫn giải

chọn C

$\begin{array}{l}y=\sqrt{\tan x+\cot x}\\ \Rightarrow y^2=\tan x+\cot x\\ \Rightarrow y\text{'}.2y=\frac{1}{{\cos }^{2}x}-\frac{1}{{\sin }^{2}x}\end{array}$

$\Rightarrow y\text{'}=\frac{1}{2\sqrt{\tan x+\cot x}}\left(\frac{1}{{\cos }^{2}x}-\frac{1}{{\sin }^{2}x}\right)$

$\begin{array}{l}f\text{'}\left(\frac{\pi }{4}\right)\\ =\frac{1}{2\sqrt{\tan \frac{\pi }{4}+\cot \frac{\pi }{4}}}\left(\frac{1}{{\cos }^2\left(\frac{\pi }{4}\right)}-\frac{1}{{\sin }^2\left(\frac{\pi }{4}\right)}\right)\\ =\frac{1}{2\sqrt{2}}\left(2-2\right)=0\end{array}$