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 ) ]
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.
Differentiate x2logx ?
We will apply the Product rule to the following equation.
= 2x log x+ x2/x ( The Derivation of log x is (1 / x),
= x( 2log x+1 ).
Differentiate log2x ?
= log x / x + log x / x,
= 2log x / x.