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What is the derivative of # arctan sqrt(x^2 -1)#?

Answer 1

Chain rule

#y = tan^(-1)(sqrt(x^2-1))# #dy/(dx) = 1/(1+(sqrt(x^2-1))^2) *1/(2sqrt(x^2-1)) . 2x# #= 1/(x.sqrt(x^2-1))#

Another way to solve this is to use different variables and then reduce the equation. Let #u = sqrt(x^2-1)#, #v=x^2#, then #y = tan^(-1)u# #u = sqrt(v-1)# #dy/(dx) = dy/(du).(du)/(dv).(dv)/(dx)# #= 1/(1+u^2).(1/(2sqrt(v-1))).2x# substituting the relationships back we get,#dy/(dx) = 1/(1+(sqrt(x^2-1))^2) *1/(2sqrt(x^2-1)) . 2x# #= 1/(x.sqrt(x^2-1))#

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Answer 2

Ezra Adams

The derivative of arctan(sqrt(x^2 - 1)) is (x) / (sqrt(x^2 - 1) * (x^2 - 1)).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.