问题及解答

求下面函数的导数

\[
y=\frac{\sqrt{x^2+1}+\sqrt{x^2-1}}{\sqrt{x^2+1}-\sqrt{x^2-1}}
\]

\[
\begin{split}
y&=\frac{\sqrt{x^2+1}+\sqrt{x^2-1}}{\sqrt{x^2+1}-\sqrt{x^2-1}}\\
&=\frac{(\sqrt{x^2+1}+\sqrt{x^2-1})^2}{(x^2+1)-(x^2-1)}\\
&=\frac{2x^2+2\sqrt{x^4-1}}{2}\\
&=x^2+\sqrt{x^4-1},
\end{split}
\]

于是

\[
y'(x)=2x+\frac{4x^3}{2\sqrt{x^4-1}}=2x+\frac{2x^3}{\sqrt{x^4-1}}.
\]