Compute the derivative of arcsin(x).

To compute the derivative of arcsin(x) we use the fact that it is the inverse of sine. Write y=arcsin(x). We want dy/dx. Taking sin on both sides yields sin(y)=x. Use implicit differentiation to differentiate both sides with respect to x. We obtain cos(y)*(dy/dx)=1 --> dy/dx=1/cos(y). Now sin(y)=x and we have the pythagorean identity sin^2(y)+cos^2(y)=1. This gives cos^2(y)=1-sin^2(y)=1-x^2 and so cos(y)=sqrt(1-x^2) (reason for choosing +sign is that cos(y)>0 on range of y=arcsin(x)). Thus dy/dx=1/sqrt(1-x^2). So derivative of arcsin(x) is 1/sqrt(1-x^2).