Expert: Paul Klarreich - 2/22/2006

Question
find the second derivative of:

x^1/2 + y^1/2 = 1  

Answer
Hi, Chevonne,

Subject:  implicit differentiation
Question:  find the second derivative of:

x^1/2 + y^1/2 = 1

Assuming you know how to do implicit differentiation in the first place, to find the second derivative:

1. Find the first derivative in the 'usual' implicit way.  

2. Take your y' and differentiate that.   Since you still have some y-s in it, you will have some y'-s in the result.

3. Where you see y', substitute the result we just got.

3.5:  Prepare a pot of black coffee and have an aspirin ready.

3. Simplify.

WARNING: THE MATERIAL BELOW MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.


For:

x^1/2 + y^1/2 = 1

1/2x^(-1/2) + 1/2y^(-1/2) y' = 0

x^(-1/2) + y^(-1/2) y' = 0

y^(-1/2) y' = - x^(-1/2)

    - x^(-1/2)   - y^1/2
y' = ---------- = --------  FIRST DERIVATIVE
     y^(-1/2)      x^1/2


OK.  Now use the quotient rule to differentiate that.

     x^1/2( 1/2y^(-1/2) y') - (y^1/2)(1/2x^(-1/2)
y'' = --------------------------------------------
                         x

(multiply through by x^1/2)

     x( 1/2y^(-1/2) y') - (y^1/2)(1/2)
y'' = ------------------------------------
                     x^3/2


(multiply through by 1/2)


     x( y^(-1/2) y') - (y^1/2)
y'' = -----------------------------
               2x^3/2


(multiply through by  y^1/2)

       x y' - y
y'' = --------------
      2x^3/2 y^1/2

NOW replace y' by the answer we got earlier.


       x (- y^1/2 / x^1/2)  - y
y'' = -------------------------
         2x^3/2 y^1/2

(multiply through by  x^1/2)

      - xy^1/2  - x^1/2 y
y'' = -------------------------
         2x^2 y^1/2

Factor:


      - x^1/2 y^1/2(x^1/2  + y^1/2)
y'' = ------------------------------
            2x^2 y^1/2

Say AHA! and replace:


      - x^1/2 y^1/2(1)
y'' = ----------------
         2x^2 y^1/2

A little more simplifying:

       - 1
y'' = --------
      2x^3/2

Well, maybe there were other ways, but...