# Derivative of ln x

so i tried to do the derivative of natural logarithm today(natural cuz e^x has simple derivative)
$$f'(x)=\lim_{dx\to 0}\frac{ln(x+dx)-lnx}{dx}$$
logarithm identity $$\log_ab+\log_ac=\log_a(b*c)$$
$$f'(x)=\lim_{dx\to 0}\frac{ln(\frac{x+dx}{x})}{dx}$$
simplify the fraction (and put 1/dx into ln cuz of log identity)
$$f'(x)=\lim_{dx\to 0}{\ln((1+\frac{dx}{x})^{\frac{1}{dx}})}$$
the thingy inside the lim is identical to the compound interest formula
$$\lim_{x\to\infty}p(1+\frac{rt}{x})^{x}$$
or the base of that spikey ln(which is derived as a limit of x->infinity from that formula)
$$e=\lim_{n\to\infty}(1+\frac{1}{n})^{n}$$
then put e into infinite compound interest formula
$$pe^{rt}$$
r and t become exponents of e
so $$\lim_{dx\to 0}{\ln((1+\frac{dx}{x})^{\frac{1}{dx}})}=\ln(e^\frac{1}{x})=\frac{1}{x}$$
(note $$lim_{x\to0}\frac{1}{x}=\infty$$ so dx=1/n and 1/x is the coeffecient of dx so it goes to exponent too)
Conclusion:the derivative of ln(x) is 1/x

Cool. I sort of wish to learn how to do this someday. I don't know what I'd use it for though.

If you work in science, engineering, data science, or math you'll use calculus all the time. You won't have to make proofs about the derivative of some function; you'll just look those up in Wolfram Alpha. But you'll have to understand what the derivative means to use it when appropriate.

WAT I THOUGHT U LEARNED THAT?!?!?!??!?!

tsiolkosvoky rocket equation etc

derivative->detect changes(a change in position->velocity)

(ps:just to be clear,learning calculus does not help you make speedometers,they just approximate the derivative and choose a large but still invisible to the human eye deltax)

well i know what a derivative is, but not how to calculate it.

uh
adding rule:$$(u\pm v)'=u'\pm v'$$
multiply rule:$$u'v'=uv'+u'v$$(?)
divide rule:(actually another form of multiply rule)$$(\frac{high}{low})'=$$low the high and high the low,over the line and square the low
power rule:$$(u^c)'=c(u')^{c-1}$$(c is constant,if not,apply exponent rule)
natural rules:exp rule $$(e^x)'=e^x$$ln rule$$(\ln x)'=\frac{1}{x'}$$

Those rules aren't really useful, because they only cover special cases and he wants to know how it works in general. So you need to show him an epsilon-delta proof.

Oh.
Delta epsilon:Limits (Formal Definition)
General formula:$$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}$$
the dx should really be delta x(the "heat this and boil"sign of chemistry or $$b^2-4ac$$)
but my katex gives $$\delta$$ which is also the wrong sign(going partial?)

This awesome guy did it too
To my surprise this guy did it almost identically to my answer
but he broke down the ln using an additional substitution step instead of directly appling experience to compound interest formula