Expert: Paul Klarreich - 10/30/2006

Question
hello~
We are learning about concavity, and in general i am having difficulty findng second derivatives ... i am horrible at math and figuring out order of operations.
For example:
           24
f(x) = -----------
       (x^2 + 12)

i can find the first derivative fairly easily (in this case using the quotient rule)

         [(0)(x^2 + 12) - (2x + 0)(24)]
f'(x) = ---------------------------------
             (x^2 + 12)^2

         [0 - 48x]                 -48x
     = ---------------    =  -----------------
         (x^2 + 12)^2           (x^2 + 12)^2

But then i do the second derivative (once again using the quotient rule) or do rewrite the equation and use the chain rule ... either way, i cant seem to get the right answer (i had a solution manual to help, but it doesnt help, because it doesn't show me step by step, it just gives the answers).

If i used the quotient rule:

       [(-48)(x^2 + 12)^2] - [(2)(x^2 + 12)(-48x)(2x)]
f"(x) = ------------------------------------------------
                    [(x^2 + 12)^2]^2

       [(-48)(x^2 + 12)^2] - [(-192x)(x^2 + 12)]
     = -------------------------------------------
                  [(x^2 + 12)^4]

        [(-48)(x^2 + 12)^2] - [(-192x^3 - 2304)]
     = -------------------------------------------
                  [(x^2 + 12)^4]

        [(-48)(x^2 + 12)^2] + 192x^3 + 2304)]
     = -------------------------------------------
                  [(x^2 + 12)^4]

... and then i am lost somehow ... if i don't factor out the -(-192x), and instead, subtract it from the (-48x), i get:     (+144)[(x^2 + 12)^2 - (x^2 + 12)]
        ----------------------------------
                [(x^2 + 12)^4]

If i rewrite the first derivative to :

    (-48x)(x^2 + 12)^-2

i still get a (+144) and the rest of the junk u saw above.

the solution books says:


         (-144)(4-x^2)
f"(x) = ------------------
         (x^2 + 12)^3

According to their solution, i understand how the direction of the convcavity can be found when the numerator is set to zero, x will equal plus or minus 2, where it would be concave upward on (negative infinity, -2) and (2, positive infinity).

I hope this wasn't too long of an explaination, and I hope you can help me make sense of all this.

Thanks,
Alison


Answer
Questioner:  alison
Category:  Calculus
Private:  no
 
Subject:  second derivative
Question:  hello
We are learning about concavity, and in general i am having difficulty findng second derivatives ... i am horrible at math and figuring out order of operations.

>> You are better than you let on.  The work here isn't as bad as you think.


For example:
          24
f(x) = -----------
      (x^2 + 12)

I can find the first derivative fairly easily (in this case using the quotient rule)

>> Actually, writing  24(x^2 + 12)^-1 is easier, but only because the numerator is a constant.  See comment at the end.


        [(0)(x^2 + 12) - (2x + 0)(24)]    << Make the first term zero NOW.
f'(x) = ---------------------------------
            (x^2 + 12)^2

        [0 - 48x]                 -48x
f'(x) = ---------------    =  -----------------
        (x^2 + 12)^2           (x^2 + 12)^2

>> Good, so far.



But then I do the second derivative (once again using the quotient rule) or do rewrite

the equation and use the chain rule

>>>  NO, NO, NO -- don't do that.  Use the QR.


... either way, i can't seem to get the right answer (i had a solution manual to help,

but it doesnt help, because it doesn't show me step by step, it just gives the answers).

If i used the quotient rule:

      [(-48)(x^2 + 12)^2] - [(2)(x^2 + 12)(-48x)(2x)]
f"(x) = ------------------------------------------------
                   [(x^2 + 12)^2]^2

      [(-48)(x^2 + 12)^2] - [(-192x)(x^2 + 12)]    <<< Get the x^2+12 out of here.
    = -------------------------------------------
                 [(x^2 + 12)^4]

       [(-48)(x^2 + 12)^2] - [(-192x^3 - 2304)]  <<< Mistake here. 2304x
    = -------------------------------------------
                 [(x^2 + 12)^4]

       [(-48)(x^2 + 12)^2] + 192x^3 + 2304)]
    = -------------------------------------------
                 [(x^2 + 12)^4]

... and then i am lost somehow ... if i don't factor out the -(-192x), and instead,

subtract it from the (-48x), i get:     (+144)[(x^2 + 12)^2 - (x^2 + 12)]
       ----------------------------------
               [(x^2 + 12)^4]

If i rewrite the first derivative to :

   (-48x)(x^2 + 12)^-2         <<< VERY BAD IDEA.(see note)

i still get a (+144) and the rest of the junk u saw above.

the solution books says:


        (-144)(4-x^2)
f"(x) = ------------------
        (x^2 + 12)^3

According to their solution, i understand how the direction of the convcavity can be found when the numerator is set to zero, x will equal plus or minus 2, where it would be concave upward on (negative infinity, -2) and (2, positive infinity).

I hope this wasn't too long of an explanation, and I hope you can help me make sense of all this.

Thanks,
Alison

.....................................................
Hi, Ali,

[[No offense meant -- my niece is Alison and everyone calls her Ali.]]

OK, let's give it a try.

                  x
f'(x) = - 48  -------------    << Remove the -48; that simplifies a bit.
             (x^2 + 12)^2

              ((x^2 + 12)^2)(1) - (x)(4x(x^2 + 12))
f''(x) = - 48  -------------------------------------
                      (x^2 + 12)^4


              (x^2 + 12)^2 - (4x^2)(x^2 + 12)  << DON'T MULTIPLY OUT HERE.
f''(x) = - 48  --------------------------------
                      (x^2 + 12)^4

Now a factor of  x^2 + 12  cancels.

              (x^2 + 12) - (4x^2)
f''(x) = - 48  --------------------
                   (x^2 + 12)^3

              x^2 + 12 - 4x^2       << Now remove parentheses.
f''(x) = - 48  --------------------
                   (x^2 + 12)^3


              12 - 3x^2
f''(x) = - 48  ------------
              (x^2 + 12)^3

              3(4 - x^2)     << Factor just a little.
f''(x) = - 48  ------------
              (x^2 + 12)^3

                4 - x^2
f''(x) = - 144  ------------
               (x^2 + 12)^3


Note about the quotient rule:

Any time you have

f(x) = u(x)/v(x), you can rewrite it as  u(x) [v(x)]^-1.

Then you don't have to use the quotient rule.  Instead, however, you will need both the product rule and the chain rule.  In my 57 years of teaching calculus, a number of students have tried to do examples this way.  

I don't remember how many tried this.
I DO remember how many of them got the examples right.  

ZERO!  

So learn to love the quotient rule.  Can't?  OK, learn to accept it and use with it.

The example earlier -- 24(x^2 + 12)^-1 is not one of these, because the first factor is a constant and so you don't need the product rule.

Actually, you didn't do badly.  
You made an error in judgment when you decided to multiply out (-192x)(x^2 + 12).  With some experience you'll avoid this.
You made an error in doing the multiplication -- you lost a factor of x in the second term.

Aside from that, you might have done OK.

.......................................
Note about 'concavity':  When your teacher says to draw these conclusions:

f'' positive -->  concave upward.
f'' negative -->  concave downward.

I recommend that you replace them with:

f'' positive -->  graph turning to the left.
f'' negative -->  graph turning to the right.

HOWEVER, these conclusions are only valid if you know how to drive a car.  'turning to the left', for example, means that if a car is driving along the graph (always going from left to right on the coordinate system) then the driver must turn the steering wheel to the left in order to stay on the graph.