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Show that if y=sinh1x, then y=ddxsinh1x=1 x2+1.

Solution Since y=sinh1x=ln(x+ x2+1), we have y=ddxsinh1x=ddx[ln(x+ x2+1)]=ddxln(u)=u(x)u(x)ddx(x+ x2+1)x+ x2+1=1+12(x2+1)1/22xx+ x2+1=1+x x2+1x+ x2+1= x2+1+x x2+1x+ x2+1=x+ x2+1(x+ x2+1) x2+1=1 x2+1