Derivative of ln (x) = 1/x
Given
(d(ex)/dx=ex
eln(x)=x
using implicit differentiation:
y = ln (x)
eln(x)=x
ey = eln(x)=x,y=ln (x)
Differentiate both sides:
ey =x,y= ln (x)
(d(ey))/dx= (d(x))/dx
ey dy/dx=1
(d(y))/dx= 1/ey ,y= ln (x) divide ey on both sides
(d(y))/dx= 1/ey = 1/eln(x) , eln(x) = x
(d(y))/dx= (d(ln(x))/dx= 1/x
