l>The Derivative the the organic Logarithm
 The Derivative the the organic LogarithmDerivation of the DerivativeOur next task is to identify what is the derivative that the naturallogarithm. We start with the station definition. If y= ln xthen ey= xNow implicitly take the derivative the both sides with respect to xremembering to main point by dy/dx top top the left handside because it is offered in regards to y not x. eydy/dx = 1From the inverse definition, we can substitute x in because that ey to get x dy/dx= 1Finally, division by x to get dy/dx= 1/xWe have proven the adhering to theorem Theorem (The Derivative of the herbal Logarithm Function) If f(x) = ln x, then f "(x) = 1/x

Examples

Find the derivative of

f(x) = ln(3x - 4)

Solution

We use the chain rule. We have

(3x- 4)" = 3

and

(lnu)" = 1/u

Putting this together gives

f "(x)= (3)(1/u)

3 =3x - 4

Example

find the derivative of

f(x)= ln<(1 + x)(1 + x2)2(1 + x3)3 >

Solution

The last thing that we want to carry out is to usage the product rule and also chain rulemultiple times. Instead, we an initial simplify through properties that the naturallogarithm. We have

ln<(1 + x)(1+ x2)2(1 + x3)3 > = ln(1+ x) + ln(1 + x2)2 + ln(1 + x3)3

= ln(1+ x) + 2 ln(1 + x2) + 3 ln(1 + x3)

Now the derivative is not so daunting. We have actually use the chain ascendancy toget

14x9x2 f "(x)=++1 + x1 + x21 + x3

Exponentials and With various other Bases
 definition allow a > 0 climate ax = ex ln a
ExamplesFind the derivative turn off (x) = 2x Solution
We create 2x = ex ln 2 currently use the chain preeminence f "(x) = (ex ln 2)(ln 2) = 2x ln 2 Logs With other BasesWe define logarithms with other bases by thechange of base formula.

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 an interpretation ln x loga x = ln a
Remark:
The nice part of this formulais that the denominator is a constant. We do not need to use the quotientrule to discover a derivativeExamples discover the derivative that the following features f(x) = log4 x f(x) = log in (3x + 4) f(x) = x log(2x) Solution We usage the formula ln x f(x) = ln 4 so that 1 f "(x) = x ln 4 us again use the formula ln(3x + 4) f(x) = ln 10 now use the chain dominance to obtain 3 f "(x) = (3x + 4) ln 10