What is d/dx of cos^-1 (1-x/1+x) ?

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Answer 1 · 2018-05-31T17:40:10

#color(blue)[d/dx[arccos((1-x)/(1+x))]=2/[(1+x)^2sqrt(1-((1-x)/(1+x))^2]]#

note that #cos^-1((1-x)/(1+x))=arccos((1-x)/(1+x))#

#d/dx[arccos(u)]=-1/sqrt(1-(u)^2)*u'#

#d/dx[arccos((1-x)/(1+x))]=-[[-(1+x)-(1-x)]/(1+x)^2]/sqrt(1-((1-x)/(1+x))^2#

#=-[-(1+x)-(1-x)]/[(1+x)^2sqrt(1-((1-x)/(1+x))^2]#

#=-[-1-x-1+x]/[(1+x)^2sqrt(1-((1-x)/(1+x))^2]#

#=-[-2]/[(1+x)^2sqrt(1-((1-x)/(1+x))^2]#

#=2/[(1+x)^2sqrt(1-((1-x)/(1+x))^2]#