How do I differentiate x with respect to y when x=√(siny+1)

Question

This is an A level question

Answer 1

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))