Calculus Derivatives and Integrals of lnX
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Loggy
Derivatives and Integrals of lnX

 

 The twin of exponential equations are logarithms.

 If we say ...

 Y = eX

 That also means ...

 log eY = X

 (also known as ln Y = X)

 
 Remember the way you read a log is:

 

 
 Which is the same thing we said in the beginning ...

 Y = eX

 

 (NOTE: From here on we'll be using ln instead of loge)

 
 So the point is:

 Y = eX    and   ln Y = X

 Say the exact same thing.
 
 So let's talk about the derivatives and integrals of things like
 

 ln X

 
 When we went looking for the derivative of eX, we found that it was eX,
 give or take a dX.
 Wouldn't it be nice if ln X worked the same way
 SORRY, No such luck.
 But finding it isn't too bad.
 
 Start with 

 Y = ln X

 
 which is the same thing as saying:

X = eY 

 (Don't worry, we'll be back to ln X before we're done)

 
 OK, the derivative of eY (with respect to the variable Y) is eY
 so we can say:

 

 
 We can flip these over and say:

 

 
 Now we do some substitution.
 We started with the equations:

 

  Y = ln X   also known as   X = eY 

 
 Now we substitute both of these into our last equation:
 

 

 
 In English, what this says is that the derivative of lnX,
 is equal to 1/X (also known as X-1)
 This also means that the integral of X-1 is lnX,
 remembering, of course, to include dX's and + C's where needed.
 

 

 
 Sometimes you will see written as
 Don't let that throw you, they both mean the same thing.
 
 Examples:

 

 
 Did you notice the 2X in the numerator of the last one?
 The derivative of X2 is 2XdX,  so everything we needed to have,
 to "Put Back Into" the 1/X2 was there.
 
 So this all works out great for things like eX and ln X,
 but what if you have to deal with 2X or log8X ?
 Huh???
 Well?????
 

   copyright 2005 Bruce Kirkpatrick

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