Question

prove that d/dx × log x =1/x​

Answer 1

Topic :-

Differentiation

To Prove :-

$\dfrac{\mathrm{d} }{\mathrm{d} x}(\log_e x)=\dfrac{1}{x}$

Solution :-

We know that,

$\dfrac{\mathrm{d} }{\mathrm{d} x}(x)=1$

We can write,

$x=e^{\log_ex}$

$\because x^{\log_ee}=e^{\log_ex}$

$\because \log_ee=1$

Replacing value of 'x',

$\dfrac{\mathrm{d} }{\mathrm{d} x}(e^{\log_ex})=1$

Differentiating by Chain Rule,

$e^{\log_ex}\dfrac{\mathrm{d} }{\mathrm{d} x}(\log_ex)=1$

$\dfrac{\mathrm{d} }{\mathrm{d} x}(\log_ex)=\dfrac{1}{e^{\log_ex}}$

$\because e^{\log_ex}=x^{\log_ee}=x$

$\therefore \dfrac{\mathrm{d} }{\mathrm{d} x}(\log_ex)=\dfrac{1}{x}$

Hence, Proved !!

Note :

There are several methods to prove the given statement.

Some common methods are Reverse Anti-Derivate Method and First Principle of Differentiation.

$\dfrac{\mathrm{d} }{\mathrm{d} x}(e^x)=e^x\:is\:used\:while\:applying\:Chain\:Rule.$