Derivative of an Exponential Function

Date: 04/23/2007 at 21:32:01
From: Bob
Subject: Derivative of an exponential function?

Hello.  I can easily find derivatives of such functions as y = x^4,
but how do you solve y = 4^x?  By going to a web site I see that the 
answer is y'= 4^x * log(4), but how does one reach that answer?  
Thanks much!

Date: 04/24/2007 at 13:05:58
From: Doctor Susie
Subject: Re: Derivative of an exponential function?

Hi Bob, 

Thanks for writing.  This is a neat problem because it is very hard
to do it as is, but by using logs we can make it a lot easier.  The
original equation is:

  y = 4^x

Now since I can't easily do the differentiation with the x in the
exponent I want to move it.  I know that:

  log(a^b) = b*log(a)

So I am going to take the log of both sides:

  log(y) = log(4^x) = x*log(4)

Now I am going to differentiate the equation.  The differential of
log(y) is 1/y.  So I get: 

  1  
  - * dy = log(4) * dx
  y

If I rearrange this a little I get:

  dy 
  -- = log(4) * y
  dx
  
now we know that y = 4^x so we can plug that in:

  dy
  -- = log(4) * 4^x
  dx

where y' = dy/dx

I hope this helps clear up where the differentiation comes from.  If
you have any more questions please feel free to drop us a line. 


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