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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'?

Thanks,
Julie
```

```
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.

Try

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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