Applying the chain rule twice

let's say that Y is equal to sine of x squared to the third power which of course we could also write as sine of x squared to the third power and will work curious about is what is the derivative of this with respect to X what is dy/dx which you can also write as Y prime well there's a couple of ways to think about it this isn't a straightforward expression here but you might notice that I have something being raised to the 3rd power in fact if we look at the outside of this expression we have some business in here it's being raised to the 3rd power and so one way to tackle this is to apply the chain rule so if we apply the chain rule it's going to be the derivative of the outside with respect to the inside or the something to the third power the derivative of the something to the third power with respect to that something so it's going to be three times that something squared times the derivative with respect to X of that something in this case this something is signed we write that in the blue color it is sine of x squared it is sine of x squared if I no matter what was inside of these orange parentheses I would put it inside of the orange parentheses and these orange brackets right over here we learned that in the chain rule so let's see we know this is just a matter the first part of the expression is just a matter of algebraic simplification but the second part we need to now take the derivative of sine of x squared well now we would want to use the chain rule again so I'm going to take the derivative its sine of something so this is going to be the derivative of this it's going to be the sine of something with respect to something so that is cosine of that something times the derivative with respect to X of the something in this case the something is our x squared and if course we have all of this out front which is the three times sine of x squared I could write it like this squared all right so we're getting close now we just to figure out the derivative with respect to X of x squared and we've seen that many times before that we just use the power rule that's going to be 2x 2x and so if we wanted to write being the dy/dx you get a little bit of a mini drumroll here this shouldn't take us too long dy/dx I'll multiply the 3 times the 2x which is going to be 6x so I covered those so far times sine squared of x squared times sine squared of x squared times cosine of x squared and we are done multiplying the chain rule multiple times