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/
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