Calculate the first derivative using the product rule:
(dz)/dx = d/dx ( xsqrt(x+1) ) = (d/dx x) sqrt(x+1) + x (d/dx sqrt(x+1))dzdx=ddx(x√x+1)=(ddxx)√x+1+x(ddx√x+1)
(dz)/dx = sqrt(x+1) + x/(2sqrt(x+1))dzdx=√x+1+x2√x+1
Simplifying:
(dz)/dx = (2(x+1) + x)/(2sqrt(x+1)) = (3x+2)/(2sqrt(x+1))dzdx=2(x+1)+x2√x+1=3x+22√x+1
Differentiate again using the quotient rule:
(d^2z)/dx^2 = d/dx ((3x+2)/(2sqrt(x+1)))d2zdx2=ddx(3x+22√x+1)
(d^2z)/dx^2 = ((2sqrt(x+1))d/dx (3x+2) - (3x+2)(d/dx 2sqrt(x+1)) )/(2sqrt(x+1))^2d2zdx2=(2√x+1)ddx(3x+2)−(3x+2)(ddx2√x+1)(2√x+1)2
(d^2z)/dx^2 = ((6sqrt(x+1)) - (3x+2)/sqrt(x+1)) /(4(x+1))d2zdx2=(6√x+1)−3x+2√x+14(x+1)
(d^2z)/dx^2 = (6x+6 - 3x-2) /(4(x+1)^(3/2))
(d^2z)/dx^2 = (3x+4) /(4(x+1)^(3/2))