How do you differentiate #y=csc^-1(x/2)#?

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Answer 1 · 2016-11-30T02:54:11

#dy/dx=(-2)/(x^2sqrt(1-4/x^2)#

#y=csc^-1(x/2)#

#cscy = x/2#

#siny = 2/x#

#cosy dy/dx = -2/x^2# (Implicit differentiation and Power rule)

#dy/dx = -2/x^2 * 1/cosy#

Since #cos^2y + sin^2y =1#

#cos^2y = 1-sin^2y = 1-(2/x)^2#

#cosy = sqrt(1-4/x^2)#

#dy/dx = (-2)/(x^2sqrt(1-4/x^2)#