How do you find the derivative of tan(arcsin(x))?

1 Answer
George C.
Aug 28, 2016

d/(dx) tan(arcsin(x))= 1/(1-x^2)^(3/2)

Explanation:

Let t = arcsin(x)

Then:

x = sin(t)

So:

tan(arcsin(x)) = tan(t) = sin(t)/cos(t) = x/sqrt(1-x^2)

So:

d/(dx) tan(arcsin(x))

= d/(dx) (x (1-x^2)^(-1/2))

= (1-x^2)^(-1/2) + x*(-1/2)(1-x^2)^(-3/2)*(-2x)

= (1-x^2)(1-x^2)^(-3/2) +x^2(1-x^2)^(-3/2)

= 1/(1-x^2)^(3/2)

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