The Derivative the the organic Logarithm Derivation of the Derivative Our next task is to identify what is the derivative that the naturallogarithm. We start with the station definition. If y= ln x then ey= x Now 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 = 1 From the inverse definition, we can substitute x in because that ey to get x dy/dx= 1 Finally, division by x to get dy/dx= 1/x We 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 |
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| an interpretation ln x loga x = ln a |