If (dy)/(dx)=sqrt(y^2+1)dydx=y2+1, then (d^2y)/(d^2x)d2yd2x is what?

I took the derivative of this function and got y/(sqrt(y^2+1))yy2+1. However, the correct answer is sqrt(2y)2y. How?

2 Answers
May 1, 2018

(d^2y)/(dx^2)=yd2ydx2=y
(not sqrt2y2y)

Explanation:

We use implicit differentiation

As (dy)/(dx)=sqrt(y^2+1)dydx=y2+1

(d^2y)/(dx^2)=1/(2sqrt(y^2+1)) * 2y * (dy)/(dx)d2ydx2=12y2+12ydydx

= 1/(2sqrt(y^2+1)) * 2y * sqrt(y^2+1)12y2+12yy2+1

= yy

May 1, 2018

=> (d^2y)/(dx^2) = y d2ydx2=y

Explanation:

=> ((dy)/(dx))^2 = y^2 + 1 (dydx)2=y2+1

=> 2 (dy)/(dx) (d^2y)/(dx^2) = 2y (dy)/(dx) 2dydxd2ydx2=2ydydx

=> cancel(2 (dy)/(dx)) (d^2y)/(dx^2) = cancel(2 (dy)/(dx))y

=> (d^2y)/(dx^2) = y