*Notes to a video lecture on http://www.unizor.com*

__Derivative Examples -__

Logarithmic Functions

Logarithmic Functions

1.

*f(x) = ln(x)*(

**is a**

*ln(x)**natural*logarithm with base

*- a fundamental constant in Calculus, approximately equal to*

**e***2.71*)

*f*

^{ I}**(x) = 1/x***Proof*

The function increment is

**Δ**

*ln(x+***=**

*x)−ln(x)*=

**Δ**

*ln[(x+***=**

*x)/x]*=

**Δ**

*ln(1+*

*x/x)*Now we can use an amazing limit

**as**

*(1+x)*^{1/x}→ e

*x→0*where

**is the same fundamental constant as above.**

*e*Based on this,

**as**

*ln[(1+x)*^{1/x}] → ln(e)

*x→0*Using the properties of logarithms, we can transform it into

**as**

*[ln(1+x)]/x → 1*

*x→0*(that is,

**is infinitesimal variable)**

*x*Let's use this property in calculation of our derivative.

**f**^{ I}**(x) =***Δ*

**= lim**_{Δx→0}**[ln(1+****Δ**

*x/x)]/*

*x =*(substitute

*Δ*

**δ=****)**

*x/x**=*

**= lim**_{δ→0}**[ln(1+δ)]/(x·δ)**

**= {lim**_{δ→0}**[ln(1+δ)]/δ}/x**As we noted above,

**as**

*[ln(1+x)]/x → 1*

*x→0*In our case the role of infinitesimal

**is played by variable**

*x→0**.*

**δ**Therefore,

**lim**_{δ→0}**[ln(1+δ)]/δ = 1**from which we conclude

**f**^{ I}**(x) = 1/x**2.

*f(x) = log*_{b}(x)

*f*

^{ I}**(x) = 1/[x·ln(b)]***Proof*

We will use the following property of logarithms that allows to change the base:

*log*

_{b}(x) = log_{c}(x)/log_{c}(b)Using this, we, firstly, convert

*log*into natural logarithm with base

_{b}(x)*:*

**e**

**log**_{b}(x) = ln(x)/ln(b)Now we see that function

**differs from function**

*log*_{b}(x)**only by a factor**

*ln(x)**.*

**1/ln(b)**Therefore, considering the expression for a derivative of

*ln(x)**,*

*f*^{ I}**(x) = 1/[x·ln(b)]**
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