題目 2. (7%)(8%)(7\%)(8\%)(7%)(8%) Find the derivatives of yyy with respect to xxx. (2a) x3+y2=sin2(y)x^3+y^2=\sin^2(y)x3+y2=sin2(y). (2b) y=tan−1(x2−1)+csc−1(x), x>1y=\tan^{-1}\left(\sqrt{x^2-1}\right)+\csc^{-1}(x),\ x>1y=tan−1(x2−1)+csc−1(x), x>1. 解答 解法一 答題過程 展開 (2a) 微分後 3x2+2yy′=2sinycosy y′3x^2+2yy'=2\sin y\cos y\,y'3x2+2yy′=2sinycosyy′，故 y′=3x2sin(2y)−2y.\boxed{y'=\frac{3x^2}{\sin(2y)-2y}}.y′=sin(2y)−2y3x2. (2b) 兩項導數分別為 1/(xx2−1)1/(x\sqrt{x^2-1})1/(xx2−1)、−1/(xx2−1)-1/(x\sqrt{x^2-1})−1/(xx2−1)，故 y′=0\boxed{y'=0}y′=0。
