 In this video, we provide the solution to question number four for practice exam number three for math 1210 In which case we're asked to compute the derivative of y equals sine inverse of the square root of sine So this function the first thing I notice is we have some composition of functions here We have the square root of sine sitting inside of Sine inverse right here. And so the chain rule is gonna be the first thing I use to calculate this derivative I see that y prime is gonna equal We have to remember the derivative of sine inverse the derivative of sine inverse of you right here It's gonna look like you prime over the square root of one minus you squared Where you in this case is the square root of square root of sine in here So we then we'll get the denominator first so we get one minus We're gonna get the square root of sine Square that'll clearly simplify But we have to also take the derivative of the square root of sine of x for which we see there with the square root of sine We're gonna use the chain rule again because we have sine of x sitting inside of the square root function So it's important to note that for the square root function if you take the derivative of the square root of u You're gonna end up with u prime over two times the square root of u So we're in this kind this case u is actually gonna be sine of x we can plug that in there So continuing on with this calculation we end up with Again the derivative the square root of sine We're gonna get the one Let me first write the outer derivative that we got from sine inverse That's gonna be one over the square root of one minus sine of x Okay, because we squared the square roots who just becomes a sine then we get the inner derivative So we have to take the derivative of one of the square root Which gives us one over two times the square root of sine Of x now we have to take the derivative of the inner function Which in this case is sine, but I know how to do that the derivative of sine is equal to cosine So it looks like our answer should be cosine over two times the square root of sine Times the square root of one minus sine Which we then see that d would then be the correct answer After we use this double application of the chain rule plus all so we need to know the derivative of sine and sine inverse