How do I differentiate x with respect to y when x=√(siny+1)
There are two ways tthat you can do this.
1) let u = siny + 1
then du/dy = cosy
x = u^(1/2)
dx/du = (1/2)u^(-1/2)
and dx/dy = (dx/du)*(du/dy)
dx/dy = (1/2)(siny + 1)^(-1/2)*(cosy)
dx/dy = (1/2)cosy/√(siny+1)
2) Let x^2 = siny + 1
differentiating both sides wrt x,
2x = cosy(dy/dx)
dx/dy = cosy/2x
dx/dy = cosy/(2√(siny+1))