A question in logarithmic differentiation

[Nanas]

In most of exercises of textbooks when ask me to differentiate f using logarithmic differentiation

some time f(x) = 0 for some values of x , so I I used logarithmic Differentiation for all x in domain of f , such that f(x) not equal zero. then prove using the definition directly that for these x the formula obtained , or prove that it doesn't exist for this zero. Is that right because most of books I had seen when give an examples of logarithmic differentiation don't care for this point I don't Know that this not important point or not.

Thanks

 

[lurflurf]

It would help if you gave specific examples. Usually this is not an important point, either the function is not differentiable at the zero or it can be recovered by limits.
consider these common examples let u and v be differentiable functions of x
we consder the quotient f'/f and can use the trivial fact that f'=f [log(f)]'
1)f=u^v
f'=u^v [log(u^v)]'=u^v [v log(u)]'=u^v [u' v/u+v' log(u)]=
in the case v=constant we easily recover f' by limit
in the case u=v f' does not exist
2)f=u v
f'=u v [log(u v)]'=u v [log(u)+log(v)]'=u v [u'/u+v'/v]->u' v+u v'
here if u or v is zero we can recover f' by limits

 

[pwsnafu]

we consder the quotient f'/f and can use the trivial fact that f'=f [log(f)]'

It's certainly not trivial, especially when f is negative.

 

[Nanas]

It would help if you gave specific examples. Usually this is not an important point, either the function is not differentiable at the zero or it can be recovered by limits.
consider these common examples let u and v be differentiable functions of x
we consder the quotient f'/f and can use the trivial fact that f'=f [log(f)]'
1)f=u^v
f'=u^v [log(u^v)]'=u^v [v log(u)]'=u^v [u' v/u+v' log(u)]=
in the case v=constant we easily recover f' by limit
in the case u=v f' does not exist
2)f=u v
f'=u v [log(u v)]'=u v [log(u)+log(v)]'=u v [u'/u+v'/v]->u' v+u v'
here if u or v is zero we can recover f' by limits

Why you didn't take absolute value first.
sorry but , What do you mean exactly by recovering by limit

 

[Nanas]

Also when we we evaluate the limit of intermediate power forms we use logarithmic differentiation but we didn't take the absolute value of the function first also it may be negative for some values in its domain.Please help me at this point also .