Differentiate Inverse Sine (arcsin) - proof

How to differentiate cos^-1x

y=cos^-1x

Bring the cos across

cosy = x

Differentiate both sides, remember when differentiating y time by dy/dx

-sin(y) dy/dx = 1

dy/dx = -1/siny

However we want to get the differential in terms of x, to do this we can use the identity

sin²t+cos²t = 1

so

sint = √(1 - cos²t)

putting this into our expression for dy/dx we get

dy/dx = 1/√(1-cos²y)

but cosy = x so

dy/dx = 1/√(1-x²)

by David Woodford