例17证明反双曲函数的导数公式 arsh x arch √x2-1 (arth x 证由 arsh x=hx+√+x2)可得 (arsh x) W x+√1+x I+x 1+x 由 arch x=l(x+√x2-1),可得( arch x)y 由athx=hn1+x,可得athx 上页 下页上页 结束 下页 例17 证明反双曲函数的导数公式 2 1 1 (arsh ) x x +  =  1 1 (arch ) 2 −  = x x  2 1 1 (arth ) x x −  =  解 由arsh ln( 1 ) 2 证 x= x+ +x 可得 2 2 2 1 1 ) 1 (1 1 1 (arsh ) x x x x x x + = +  + + +  =  2 2 2 1 1 ) 1 (1 1 1 (arsh ) x x x x x x + = +  + + +  =  2 2 2 1 1 ) 1 (1 1 1 (arsh ) x x x x x x + = +  + + +  =  由arch ln( 1) 2 x= x+ x −  可得 1 1 (arch ) 2 −  = x 由arch ln( 1) x  2 x= x+ x −  可得 1 1 (arch ) 2 −  = x x  由 x x x − + = 1 1 ln 2 1 arth  可得 2 1 1 (arth ) x x − 由  =  x x x − + = 1 1 ln 2 1 arth  可得 2 1 1 (arth ) x x −  =  结束