Find the second order derivative of y= arcsinhx ?

Answer 1

y =sinh^-1 (x)

dy/dx = d/dx(sinh^-1 (x))

         =1/√(1+x²)

Again take derivative with respect to x

d²y/dx² =d/dx(1/√(1+x²))

           =d/dx((1+x²)^(-1/2))

Formula :d/dx(x^n) =nx^n-1

  d²y/dx² =((-1/2)(1+x²)^(-3/2))d/dx(1+x^2)

              =((-1/2)(1+x²)^(-3/2))(2x)

                   =-(1+x²)^(-3/2))(x)

                    =-x/(1+x²)^(3/2)

Second derivative of sinh^-1 (x) is -x/(1+x²)^(3/2)