How I can solve this problem?

#d/(dx)(ln(x-(x^2-1)^(1/2)))=?#

1 Answer
Dec 8, 2017

The answer is #=-1/sqrt(x^2-1)#

Explanation:

Let #f(x)=ln(x-sqrt(x^2-1))#

Apply the chain rule

The derivative of #lnu(x)# is #(u'(x))/(u(x))#

#ln(u(x))'=(u'(x))/(u(x))#

#(sqrtv(x))'=(v'(x))/(2(sqrtv(x)))#

Therefore,

#f'(x)=1/((x-sqrt(x^2-1)))*(1-1/(2sqrt(x^2-1))*2x)#

#=1/((x-sqrt(x^2-1)))*(1-x/sqrt(x^2-1))#

#=1/((x-sqrt(x^2-1)))*(sqrt(x^2-1)-x)/sqrt(x^2-1)#

#=-1/sqrt(x^2-1)#