The Derivative of the logarithmic function for a variable: Here we are proving the derivative of the log function by using limits:

log ( x ) = lim _{( q} _{→} _{0 )}[ ln( x + q ) – ln( x ) ] / q = lim_{( q → 0 )} ln (( x + q ) / x ) / q

= lim _{( 1 / q )} ln ( 1 + q / x ) = lim [ ln( 1 + q/ x )^( 1 / q ) ]

Set,

p= q / x and also substitute:

lim_{( p} _{→} _{0 )}[ ln ( 1 + p )^( 1 / px )) ] = 1 / x ln [ lim _{( p} _{→} _{0 )}( 1 + p )^( 1 / p ) ]

= 1 / x ln (e) (Now in the next step we can substitute ln(e) with the value 1 )

= 1 / x.

We can use the formula in the standard form, and whenever we have differentiation of log(x) we can directly put ( 1 / x ) there. It is also interesting to see what happens when we apply the anti derivative to the derived form that is to (1 / x ) what do we get. The anti derivative of ( 1 / x ) is log(x) as well.

Hence, derivative of log x is 1/x.

Let us look at one of the derivative examples in which we use derivative log x.

Example1:

Differentiate x^{2}logx ?

We will apply the Product rule to the following equation.

= 2x log x+ x^{2}/x ( The Derivation of log x is (1 / x),

= x( 2log x+1 ).

Example2:

Differentiate log^{2}x ?

= log x / x + log x / x,

= 2log x / x.