Applying the chain rule twice

- [Instructor] Let's say that Y is equal to sin of X squared to the third power, which of course we could also write as sin of X squared to the third power and what we're curious about is what is the derivative of this with respect to X? What is DY/DX which we could 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 third power, in fact, if we look at the outside of this expression we have some business in here that's being raised to the third power. And so, one way to tackle this is to apply the chain rule. So, if we apply the chain rule it's gonna 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, the something is sin, let me write that in the blue color, it is sin of X squared. It is sin of X squared. 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 of 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 sin of X squared. Well, now we would want to use the chain rule again. So, I'm going to take the derivative, it's sin of something, so this is going to be, the derivative of this is gonna be the sin 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 of course, we have all of this out front which is the three times sin of X squared, I could write it like this, squared. Alright, so we're getting close. Now we just have 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 two X. Two X and so, if we wanted to write the DY/DX, let me get a little bit of a mini drum roll here, this shouldn't take us too long, DY/DX, I'll multiply the three times the two X which is going to be six X, so I've covered those so far times sin squared of X squared, times sin squared of X squared, times cosine of X squared. And we are done applying the chain rule multiple times.