How do you differentiate g(x) = sqrtarcsin(x^2-1) ?

1 Answer
Cesareo R.
May 24, 2016

(dy)/(dx) =x/(sqrt(arcsin(x^2-1))sqrt[1 - (1 - x^2)^2])

Explanation:

Define y = sqrt(arcsin(x^2-1)) and make
y^2=arcsin(x^2-1)->x^2-1=sin(y^2). Now following with
f(x,y)=x^2-sin(y^2) = 1
We know that (dy)/(dx) = -f_x/(f_y) = (2x)/(2y cos(y^2)). Now, substituting for y = sqrt(arcsin(x^2-1)) we have
(dy)/(dx) = x/(sqrt(arcsin(x^2-1))cos(arcsin(x^2-1))
Simplifying
(dy)/(dx) =x/(sqrt(arcsin(x^2-1))sqrt[1 - (1 - x^2)^2])

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