Calculate the Derivative for Sqrt(x^5)

Date: 03/09/2003 at 14:24:17
From: Rob
Subject: Derivatives

How do I calculate the derivative for the square root of (x to the 
fifth)?

I can do the derivative of (sqrt of x) to the fifth, but not the 
other way around. 

I can get the answer using the calculator: (5*x^4)/2*sqr rt(x^5). 
I know the expression sqrt(x^5) is the same as (x^5)^(1/2), which 
should be x^(5/2), which would mean the derivative is (5/2)x^(3/2), 
but this is not right.

Date: 03/10/2003 at 10:56:26
From: Doctor Ian
Subject: Re: Derivatives

Hi Rob,

You're correct that

  (sqrt(x))^5 = (x^(1/2))^5

              = x^(5/2)

And in fact, the derivative _is_ (5/2)x^(3/2):

  d/dx x^(5/2) = (5/2)x(5/2 - 1)

               = (5/2)x^(3/2)

To see that this is the same as the answer you got from the 
calculator, we can set them equal to each other, and simplify:

                       5x^4
     (5/2)x^(3/2) = -----------         
                    2 sqrt(x^5)


                      5x^4
         5x^(3/2) = -----------        
                     sqrt(x^5)


                       x^4
          x^(3/2) = -----------        
                     sqrt(x^5)


                       x^4
          x^(3/2) = -----------        
                      x^(5/2)


   x^(3/2)x^(5/2) = x^4


          x^(8/2) = x^4


              x^4 = x^4

This statement is true, so the expressions we started with are equal. 

So how did the calculator come up with such a complicated answer? 
Instead of simplifying, it applied the chain rule.  Letting 

  u(x) = x^5, 

this gives us 

  d/dx  sqrt(u) = (1/2)u^(-1/2) du/dx

                = (1/2)u^(-1/2) 5x^4

                    5x^4
                = ---------
                  2 sqrt(u)

                    5x^4
                = ------------
                  2 sqrt(x^5)

This, by the way, is a good illustration of why many teachers feel
that students are becoming too dependent on calculators. Your
approach was much more elegant than the calculator's, and obviously
correct, but when the calculator gave you an answer that looked
different than yours, you assumed that yours must be wrong.  

The truth is, calculators are very fast, and very accurate, but
they're not all that bright:

   Are Calculators Smart?
   http://mathforum.org/library/drmath/view/57030.html 

I hope this helps. Write back if you'd like to talk more about this,
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/