Question

d/dx log | x |=......(x≠0),Select Proper option from the given options.
(a) 1/ | x |
(b) 1/x
(c) does not exist
(d) eˣ

Answer 1

we know, any function, f(x) is differentiable at x = a only when, $\displaystyle\lim_{h\to 0^+}\frac{f(x+h)-f(x)}{h}=\displaystyle\lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$

here, x = 0 is a suspecious point. so, we have to check differentiablity at x = 0.

case 1 :- when x > 0
|x| = x
then, d/dx log|x| = d/dx logx = d(logx)/dx
= 1/x

case 2 :- when x < 0
|x| = -x
then, d/dx log|x| = d/dx log(-x) = dlog(-x)/dx
= -1/x × (-1) = 1/x

here, we see , f'(0^+) = f'(0^-) = 1/x

hence, function is differentiable and derivative of function is 1/x. so, option (b) is correct