prove that `(d(sec^(-1)x))/(dx) =1/(|x|(sqrt(x^2-1)))`

1.	prove that `(d(sec^(-1)x))/(dx) =1/(\|x\|(sqrt(x^2-1)))`
Answer» Let `y=sec^(-1)x`, where `x in R-[-1,1]` and `y in [0,pi]-{(pi)/(2)}`. Then, `x = secy` `rArr (dx)/(dy)= secy tan y gt 0` `rArr(dy)/(dx)=(1)/(secy tany)=(1)/(secy.sqrt(sec^(2)y-1))` `rArr(dy)/(dx)=(1)/(\|x\|sqrt(x^(2)-1))` Hence, `(d)/(dx)(sec^(-1)x)=(1)/(\|x\|sqrt(x^(2)-1)).`

Discussion

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