Question

$find \: the \: derivative \: of \: standard \\ function \: log_{e}(x) \\ by \: ab \: initio.$
$thank \: you$
:)

Answer 1

Hi,

Here is your answer,

Given, 

           loge(x)

y = log x/log e

y = 1/log e × log x

dy/dx = 1/xloge

So, therefore 1/xloge = 1/x (or) 1

Hope it helps you !


 

Answer 2

$\underline \frak {Hello \: there }$
$let \: y = ln_e(x ) \\ \\ then \: \frac{d}{dx} ( log_{e}(x) ) \\ \\ \Rightarrow \: \frac{d}{dx} ( log_{e}x) . \frac{d}{dx} (ex) \\ \\ \Rightarrow \: \frac{1}{ex} .e. \frac{d}{dx}(x) \\ \\ \Rightarrow \: \frac{ {e}^{ - 1} \times e}{x} \\ \\ \Rightarrow \ \boxed {\bf \frac{1}{x} }$


Hope this helps you ☺