Cho hàm số y=f(x) = cosx/(1-sinx) giá trị biểu thức

* Hướng dẫn giải

Chọn A

$\begin{array}{l}{f}^{\text{'}}\left(x\right)=\frac{{\left(\cos x\right)}^{\text{'}}\left(1-\mathrm{s}\text{in}x\right)-(1-\mathrm{s}\text{in}x{)}^{\text{'}}cosx}{{\left(1-\mathrm{s}\text{in}x\right)}^2}\\ =\frac{-\sin x.(1-\mathrm{s}\text{inx})+c\text{osx.cosx}}{{(1-\mathrm{s}\text{inx})}^2}\\ =\frac{-\sin x+{\sin }^{2}x+c{\text{os}}^{2}x}{{(1-\mathrm{s}\text{inx})}^2}\\ =\frac{-\mathrm{s}\text{inx}+1}{{(1-\mathrm{s}\text{inx})}^2}=\frac{1}{1-\mathrm{s}\text{in}x}\\ \Rightarrow f^{\text{'}}\left(\frac{\pi }{6}\right)-f^{\text{'}}\left(-\frac{\pi }{6}\right)\\ =\frac{1}{1-\frac{1}{2}}-\frac{1}{1+\text{\hspace{0.17em}\hspace{0.17em}}\frac{1}{2}}=\frac{4}{3}\end{array}$