The proof of derivative of lnx?

In summary, the proof of the derivative of ln x involves taking the limit of ln (1 + Δx/x)^(x/Δx) as Δx approaches 0. This leads to a contradiction when trying to prove the limit of (1 + 1/n)^n, as it is defined as the number that makes the base log have a derivative of 1/x. Alternatively, the proof can be done using the binomial theorem, which shows that e is the limit of a series.
  • #1
coki2000
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The proof of derivative of lnx??

Hi all,
In proof of derivative of lnx,

[tex]\frac{d}{dx}(lnx)=\lim_{\Delta x\rightarrow 0}\frac{\ln(x+\Delta x)-lnx }{\Delta x}= \lim_{\Delta x\rightarrow 0}\frac{1}{\Delta x}\ln(1+\frac{\Delta x}{x})=\lim_{\Delta x\rightarrow 0}\frac{x}{\Delta x}\frac{1}{x}\ln(1+\frac{\Delta x}{x})=\lim_{\Delta x\rightarrow 0}\frac{1}{x}ln(1+\frac{\Delta x}{x})^{(\frac{x}{\Delta x})}[/tex]

For [tex]\frac{x}{\Delta x}=n[/tex]
[tex]\lim_{\Delta x\rightarrow 0}ln(1+\frac{\Delta x}{x})^{(\frac{x}{\Delta x})}\Rightarrow \lim_{n\rightarrow\infty}\ln (1+\frac{1}{n})^{n}[/tex]

So
[tex](lnx)'=\frac{1}{x}ln(\lim_{n\rightarrow\infty}(1+\frac{1}{n})^{n})=\frac{1}{x}[/tex]

so we take the [tex]\lim_{n\rightarrow\infty}(1+\frac{1}{n})^{n}=e[/tex]

But in proof of [tex]\lim_{n\rightarrow\infty}(1+\frac{1}{n})^{n}[/tex] we take the derivative of lnx is [tex]\frac{1}{x}[/tex]
(https://www.physicsforums.com/showthread.php?t=348071)

Please explain to me this contradiction.
Thanks.
 
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  • #2


Do you mean the circular logic problem? e is defined as the number that makes the base log such that (log x)' = 1/x.

Alternatively, you could do a proof by the binomial theorem, and take the limit, you'll see the definition for e in series form.
 
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What is the proof of derivative of lnx?

The proof of derivative of lnx is based on the fundamental rule of differentiation, which states that the derivative of a function f(x) is equal to the limit of (f(x + h) - f(x)) / h as h approaches 0. By applying this rule to the natural logarithm function, we can find the derivative of lnx to be 1/x.

Why is the proof of derivative of lnx important?

The proof of derivative of lnx is important because it allows us to calculate the rate of change of natural logarithm functions. This is useful in many real-world applications, such as calculating growth rates, compound interest, and exponential decay.

What are the steps involved in proving the derivative of lnx?

The steps involved in proving the derivative of lnx include using the limit definition of derivative, simplifying the expression using logarithmic properties, and applying the limit as h approaches 0 to obtain the final result of 1/x.

Can the proof of derivative of lnx be extended to other logarithmic functions?

Yes, the proof of derivative of lnx can be extended to other logarithmic functions such as loga(x) where a is any positive real number. The same steps can be followed, but the final result will be different depending on the base a.

Are there any real-life applications of the proof of derivative of lnx?

Yes, the proof of derivative of lnx has various real-life applications in fields such as finance, economics, and physics. It is used to model natural growth and decay phenomena, as well as to calculate rates of change in various processes.

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