How do you differentiate #f(x)=ln(sqrt((x-6)/(x+6))) #?

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Answer 1 · 2015-12-06T04:17:47

#6/(x^2-36)#

#f(x)=ln(sqrt((x-6)/(x+6)))=1/2(ln(x-6)-ln(x+6))#, since:

#{(log_a(b^c)=clog_ab),(log_a(b/c)=log_ab-log_ac):}#

Use the chain rule and the rule that #d/dx[ln(u)]=(u')/u# to find the derivative:

#f'(x)=1/2((d/dx[x-6])/(x-6)-(d/dx[x+6])/(x+6))#

#=1/2(1/(x-6)-1/(x+6))#

#=1/2(((x+6)-(x-6))/((x+6)(x-6)))#

#1/2(12/(x^2-36))#

#=6/(x^2-36)#