Given x = 3sin(y/2), find dy/dx in terms of x, simplifying your answer.

The first step is to find dx/dy in terms of y, which when differentiating comes out as 3/2cos(y/2), so dy/dx in terms of y is the reciprocal of this.The next step is to eliminate the y dependent terms, which can be done one of two ways. One posssible method is to draw a diagram of a right angled triangle with an angle representing y/2 and using the relationship x = 3sin(y/2) to find cos(y/2) in terms of x using pythagoras and basic trigonometry. The other method that could be used is to utilise the trigonometric identity sin²(y/2) + cos²(y/2) = 1 and using 3sin(y/2) = x to find an expression for cos(y/2) in terms of x.Either method will give the same answer, the relationship cos(y/2) = 1/3(9-x²)^1/2. The final step is then to substitute this into dy/dx to eliminate cos(y/2) and the final expression is then dy/dx = 2/(9-x²)^1/2.