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Non-Constant Functions

Date: 02/12/2001 at 09:08:57
From: Julie Bishop
Subject: Derivative of non-constant functions

Are there any non-constant functions f and g such that (fg)' = f'g'?


Date: 02/12/2001 at 12:14:09
From: Doctor Rob
Subject: Re: Derivative of non-constant functions

Thanks for writing to Ask Dr. Math, Julie.


     f(x) = g(x) = e^(2*x)

More generally, try

     f(x) = a*e^(b*x)
     g(x) = c*e^(b*x/[b-1])

where a, b, c are nonzero and b is unequal to 1.

There are other solutions, too:

          (f*g)' = f'*g'
     f*g' + f'*g = f'*g'
     g'/g + f'/f = (g'/g)*(f'/f)
            g'/g = f'/(f'-f)

Now the integral of the left-hand side is ln|g|. If you can integrate 
the right-hand side, you can solve for g. For example, if 
f(x) = a*(x+1), then f'(x) = a, and

     ln|g(x)| = INT[-1/x dx]
              = -ln|x| + c
       |g(x)| = C/|x|
         g(x) = C/|x|

for any constant C. You can check that this pair satisfy the equation 
by using the facts that

     |x|  = sign(x)*x
     |x|' = sign(x)   except at x = 0

The first example above is obtained by letting f'/f = g'/g, which 
forced f'/f = 2, f(x) = a*e^(2*x), for any constant a.

The second example was found by setting f'/f = b for constant b.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Calculus

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