l>The Derivative of the Natural Logarithm
 The Derivative of the Natural LogarithmDerivation of the DerivativeOur next task is to determine what is the derivative of the naturallogarithm. We begin with the inverse definition. If y= ln xthen ey= xNow implicitly take the derivative of both sides with respect to xremembering to multiply by dy/dx on the left handside since it is given in terms of y not x. eydy/dx = 1From the inverse definition, we can substitute x in for ey to get x dy/dx= 1Finally, divide by x to get dy/dx= 1/xWe have proven the following theorem Theorem (The Derivative of the Natural 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 do is to use the product rule and chain rulemultiple times. Instead, we first simplify with properties of 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 use the chain rule toget

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

Exponentials and With Other Bases
 Definition Let a > 0 then ax = ex ln a
ExamplesFind the derivative off (x) = 2x Solution
We write 2x = ex ln 2 Now 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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 Definition ln x loga x = ln a
Remark:
The nice part of this formulais that the denominator is a constant. We do not have to use the quotientrule to find a derivativeExamples Find the derivative of the following features f(x) = log4 x f(x) = log (3x + 4) f(x) = x log(2x) Solution We use the formula ln x f(x) = ln 4 so the 1 f "(x) = x ln 4 We again use the formula ln(3x + 4) f(x) = ln 10 now use the chain rule to gain 3 f "(x) = (3x + 4) ln 10