Cho hàm số \(y=f\left(x\right)=\frac{1}{2}\ln\left(1+x\right)-\frac{1}{4}\ln\left(1+x^2\right)-\frac{1}{2\left(1+x\right)}\)
Tính \(f'\left(1\right)\).
\(\frac{1}{2}\)\(\frac{1}{4}\)\(\frac{1}{8}\)\(\frac{1}{12}\)Hướng dẫn giải:\(f\left(x\right)=\dfrac{1}{2}\ln\left(1+x\right)-\dfrac{1}{4}\ln\left(1+x^2\right)-\dfrac{1}{2\left(x+1\right)}\),
\(f'\left(x\right)=\dfrac{1}{2}.\dfrac{1}{1+x}-\dfrac{1}{4}.\dfrac{2x}{1+x^2}+\dfrac{1}{2\left(1+x\right)^2}=\dfrac{1}{2}\left(\dfrac{1}{1+x}-\dfrac{1}{1+x^2}+\dfrac{1}{\left(1+x\right)^2}\right)\)
\(\Rightarrow\) \(f'\left(1\right)=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{2}+\dfrac{1}{4}\right)=\dfrac{1}{8}\)