The Derivative of the Natural Logarithm
Our next task is to determine what is the derivative of the naturallogarithm. We begin with the inverse definition. If y= ln x then ey= x Now 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 = 1 From the inverse definition, we can substitute x in for ey to get x dy/dx= 1 Finally, divide by x to get dy/dx= 1/x We have proven the following theorem
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 **

**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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Remark:The nice part of this formulais that the denominator is a constant. We do not have to use the quotientrule to find a derivative

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Definition ln x loga x = ln a |

**Examples 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