Supposing y = arcsin(x), find dy/dx

Suppose:
y = arcsin(x)
Then, x = sin(y)
And, dx/dy = cos(y) ----- (1)
Using: dy/dx = 1/(dx/dy);
Thus 1 becomes: dy/dx = 1/cos(y) ------ (2)
Using: sin^2(y) + cos^2(y) = 1;
We can rearrange 2 to: dy/dx = 1/sqrt(1 - sin^2(y))
Therefore dy/dx = 1/(sqrt(1 - x^2)