Show that if y=sinh−1x, then y′=ddxsinh−1x=1 √x2+1.
Solution Since y=sinh−1x=ln(x+ √x2+1), we have y′=ddxsinh−1x=ddx[ln(x+ √x2+1)]=↑ddxln(u)=u′(x)u(x)ddx(x+ √x2+1)x+ √x2+1=1+12(x2+1)−1/2⋅2xx+ √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