Calculus Exponent Bases Other Than e
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nX
Exponent Bases Other Than e

 

 These are a bit more involved than exponents with e as a base,

 but not THAT much.
 
 The derivative of 2 X is 2 X ln X.
 
 The general rule is:

 

 
 Here's why, start with:

 Y = aX

 Take the log (ln that is) of both sides:

 ln Y = ln aX

 Use the log with exponents trick:

ln Y = X ln a

 Now solve for X:

 

 
 ln a is a constant. ln Y is a variable.
 So it's easier to see how this will work, we will separate the two.
 

 

 
 Now we take the derivative of X with respect to Y.

 

 
 WHAT? Where did that 1/Y come from?
 Don't you remember the last page?
 The derivative of ln Y is 1/Y.

 

 
 Our original problem was Y = aX, so substitute that in for Y ...

 

 

 
 But we wanted dY/dX not dX/dY!
 
 OK, so flip it over.

 

 
 The original equation was Y = aX. That means that if:

 

 

 And there it is!
 

   copyright 2005 Bruce Kirkpatrick

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