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Question

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

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Answer 1 · 2017-12-08T08:22:54

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

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

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)#