# X Derivative Ln – What Is The Derivative Of X^(Lnx)

**X Derivative Ln – What Is The Derivative Of X^(Lnx)**in

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l>The Derivative of the Natural Logarithm

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.

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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 rule f “(x) = (ex ln 2)(ln 2) = 2x ln 2 Logs With Other BasesWe define logarithms with other bases by thechange of base formula. **

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 derivative**Examples Find the derivative of the following functions 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 that 1 f “(x) = x ln 4 We again use the formula ln(3x + 4) f(x) = ln 10 now use the chain rule to get 3 f “(x) = (3x + 4) ln 10

**Derivative**

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