Tính đạo hàm của hàm số \(y=\sqrt[m+n]{\left(1-x\right)^m\left(1+x\right)^n}\) .
\(\dfrac{\left(m+n\right)+\left(m+n\right)x}{\left(m+n\right)\sqrt[m+n]{\left(1-x\right)^n\left(1+x\right)^m}}\).\(\dfrac{\left(m+n\right)-\left(m-n\right)x}{\left(m+n\right)\sqrt[m+n]{\left(1-x\right)^n\left(1+x\right)^m}}\).\(\dfrac{\left(m-n\right)-\left(m+n\right)x}{\left(m+n\right)\sqrt[m+n]{\left(1-x\right)^n\left(1+x\right)^m}}\).\(\dfrac{\left(m+n\right)+\left(m-n\right)x}{\sqrt[m+n]{\left(1-x\right)^n\left(1+x\right)^m}}\).Hướng dẫn giải:Ta có
\(y=\left(1-x\right)^{\dfrac{m}{m+n}}\left(1+x\right)^{\dfrac{n}{m+n}}\Rightarrow y'=\dfrac{-m}{m+n}\left(1-x\right)^{\dfrac{m}{m+n}-1}\left(1+x\right)^{\dfrac{n}{m+n}}+\left(1-x\right)^{\dfrac{m}{m+n}}.\dfrac{n}{m+n}\left(1+x\right)^{\dfrac{n}{m+n}-1}\)
\(y'=-\dfrac{m}{m+n}\left(1-x\right)^{-\dfrac{n}{m+n}}\left(1+x\right)^{\dfrac{n}{m+n}}+\dfrac{n}{m+n}\left(1-x\right)^{\dfrac{m}{m+n}}\left(1+x\right)^{-\dfrac{m}{m+n}}\)
\(=-\dfrac{m}{m+n}\left(\dfrac{1+x}{1-x}\right)^{\dfrac{n}{m+n}}+\dfrac{n}{m+n}\left(\dfrac{1-x}{1+x}\right)^{\dfrac{m}{m+n}}\)\(=-\dfrac{m\sqrt[m+n]{\left(1+x\right)^n}}{\left(m+n\right)\sqrt[m+n]{\left(1-x\right)^n}}+\dfrac{n\sqrt[m+n]{\left(1-x\right)^m}}{\left(m+n\right)\sqrt[m+n]{\left(1+x\right)^m}}\)
\(=\dfrac{-m\left(1+x\right)+n\left(1-x\right)}{\left(m+n\right)\sqrt[m+n]{\left(1-x\right)^n\left(1+x\right)^m}}=\dfrac{\left(n-m\right)-\left(m+n\right)x}{\left(m+n\right)\sqrt[m+n]{\left(1-x\right)^n\left(1+x\right)^m}}\)