AJ Dhiman

[mathsmajor012]

i think spending winter nights alone is one of the most horrible thing.

[mathsmajor012]

tu,tu hai wahi dil ne jise apna kaha

[mathsmajor012]

Why d/dx(lnx)=1/x?(by definition of derivative)

Let f(x)=lnx

∴ f'(x)=lim h → 0 f ( x + h ) − f ( x )/h  

∴ f'(x)=lim h→0 ln(x+h)-lnx/h

=lim h→0 ln(x+h/x)/h

=lim h→0 1/h ln(1+h/x)

=lim h→0 ln(1+h/x)^1/h

Now, ln(lim h→0 (1+h/x)^1/h

NOTE: lim t→∞ (1+1/t)^t=e

put 1/t=u

⇒ t=1/u

∴ as t→∞, u→0

therefore lim u→0 (1+u)^1/u=e

⇒ put h/x=u

as h→0,u→0

therefore ln(lim u→0(1+u)^1/ux

ln(lim u→0{(1+u)^1/u}^1/x

ln e^1/x

1/x

therefore d/dx(lnx)=1/x.