 Let's find the derivatives of the following trigonometric functions consider the function y equals sine of 6x Now this is a composition of two functions. We have one function inside of another the inner function here It's going to be 6x this it sits inside of the sine function or written slightly differently We get sine of u composed with 6x Now as we put 6x inside of our function right there That's where we get this the reason I'm decomposing this right here is to emphasize that the chain rule is going to come into play When you calculate the derivative of these functions, which the chain rule has two parts you have your dy over du your outer derivative Multiplied by the inner derivative du over dx So we have to take the derivative of the outside function with respect to the inner Variable and then take the derivative the inside function as well. So when you take the derivative here of y prime So let's be specific. We're taking the derivative of d y a derivative y with respect to x This is going to be the outer derivative first So we have to take derivative of sine the derivative of sine is in fact cosine, right? We've seen this before which the inner function u of course is a 6x So we get that we're going to get the derivative sine of u is cosine of u, but you itself is 6x It's just a it's just the inner variable we're using right there Then we have the inner derivative we have to take the derivative of 6x in this situation So we see we end up with a 6 or cosine of 6x and then the derivative of 6x with respect to x is 6 itself a Good mannered human being always puts their coefficients in front of the trigonometric function And so we see the derivative here is going to be 6 cosine of 6x Let's look at another example Let's consider this time the function f of x equals sine of x squared for which then we can actually see the Interfunction is x squared and then the outer function is again sine So we have if we were to decompose this we have the outer function is sine And we see that the inner function is x squared. We put x squared inside of sine of x So calculating the derivative here f prime of x we're going to take the outer derivative So again the derivative of sine is cosine So we get cosine of the inner function Which is x squared and then we times that by the inner derivative Which in this case the inner derivative would be a 2x take the derivative of x squared Which is to say that the derivative of our function is 2x times cosine of x squared another example Compare this one here g of x equals sine squared of x now There's a very important distinction here in this situation The inner function is actually this time sine and the outer function is going to be the squared function Written a little bit differently We see that the outer function is the squaring function u squared and the inner function is going to be sine of x So if you change the order of operations that has a big difference when it comes to function composition You know putting your socks on then your shoes is a very different process than putting your shoes on and then your socks And so this will affect the derivative in a dramatic way We see that the derivative here derivative of g with respect to x this will look like well The outer function the outer derivative is going to be two times here the inner function sine of x so notice here we take the derivative of U squared with respect to you you end up with a 2u and then we plug inside of you the Interfunction which in this case is sine of x and then we're going to times that by the inner derivative We have to take the derivative of sine with respect to x so this end up this end up giving us two sine of x Times the derivative of sine of x as we've already seen is cosine of x So the derivative would be two sine of x cosine of x and if your trig identity sense is tingling right now That's appropriate by the double angle identity. We could rewrite this as sine of 2x That last step of course is not necessary, but just simplify it of course Now compare these two right the derivative of sine of x squared was very different than the derivative of sine squared of x The order operations is critical. We have to make sure we get them in the correct order so that we get the correct derivative here All right, let's consider this one We want to take the derivative of y when it's 5 sine of 9x squared plus 2 plus cosine of pi 7s Well, since there's a sum of two things the derivative here y y prime You're going to take the derivative of 5 which since that's a constant multiple you can take the 5 out So we have to take the root of 5 or 5 times the derivative of sine of 9x squared plus 2 And then we have to take the derivative of cosine of Pi over 7 here For which when you look at the sine of 9x squared plus 2 same type of chain rule comes into play here We have as an inner function this polynomial 9x squared plus 2 the outer function is the trigonometric function sine And so when you continue with the derivative here, you're going to get five times You have to do the outer derivative which the derivative of sine is going to be cosine put the inner derivative inside of it That's why we call it the inner sorry put the inner function side That's what's the inner functions inside and then so you have your outer derivative That's important to have here your outer derivative But you have to also multiply it by the inner derivative the derivative of 9x squared plus 2 is going to be 18 x By the usual power rule. This is a critical thing to remember when you use when you calculate a derivative using the Chain rule when people take derivatives using the chain rule They nearly always remember the outer derivative no problem there Oftentimes the inner derivatives is forgotten and so much like you shouldn't forget about your inner child Otherwise you might have you know psychological problems as you grow up You shouldn't forget about your inner derivative either. Otherwise you'll have of course problems later on in this in this series, right? So that's how we can take the derivative of the first part using the chain rule The next part doesn't actually use the chain rule But I had to throw it in there because this is a this is also a common mistake the students make when you take the derivative of Cosine of pi over 7 we're so accustomed to saying things like oh the derivative of cosine is negative sign That we sometimes confuse what we're saying and what we actually need to do the derivative of cosine is not actually negative sign The derivative of cosine of x is equal to negative sign of x, right? It's important that it be a function that has to be a variable in play Drivetives are about rates of change if there's no variable. There's no change Which is why constants like the derivative of 7 here the derivative of 7 is 0 when you take the derivative of constant Doesn't matter. It's gonna be 0 or y 7. What if my constant is pi over 7? If I take the derivative of pi over 7 with respect to x that still would be 0 because pi over 7 doesn't change Well, what about this one right here? What if your constant is cosine of pi over 7? That's still gonna be 0. So the thing I'm pointing out here is that the derivative of cosine of pi over 7 is 0 It's a constant. I don't care how flamboyant that constant might seem sure I see a Commonly understood function cosine, but you've evaluated cosine in a specific number So cosine of a specific number is again a specific number. It's a constant. It's derivative will be 0 You're not gonna get anything else. Just the derivative of cosine of pi over 7 would be 0 So with that observation then out of the way, we note that 5 times 18 is equal to 90 So we get 90 x times cosine of 9 x squared plus 2 Which is then the derivative of our function here Consequence of the chain rule. Let's look at a couple more examples Let's take a look at this one right here f of x equals cosine to the fourth power of x Let's recognize. What's the inner function? What's the outer function? The inner function here is going to be cosine of x the outer function is going to be the fourth power And so with that decomposition we see that f prime of x we're going to get as the outer derivative Well, the fourth power is in place. So you're gonna get four cosine of x Cubed and then you're gonna times that by the inner derivative, which is the derivative of cosine So putting this together, of course, we get four times cosine cubed of x again When you have a when you have a exponent rate for for a trigonometric function It's customary to put it between the trigonometric function in the variable there the angle so it's cosine cubed of x and then the derivative of Cosine as we've seen before is negative sine of x like so we should probably stick that negative in front so that we don't erroneously Make it look like subtraction later on so we end up with a negative for cosine cubed x sine of x Which maybe there's a trig identity in play there. Oh, well, don't need it And then one last example same basic idea here if we take the derivative of cotangent to the sixth power recognizing that the inner function is the trigonometric function cotangent and then the outer function is The sixth power of the power function there We see that the derivative of y with respect to x here is going to be six cotangent to the fifth power x times that by the inner derivative Which the derivative of cotangent is going to be negative cosecant square root of x again Same thing as before probably stick the negative sign out in front so it doesn't look like subtraction by mistake So you get negative six cotangent to the fifth power of x times cosecant Square root of x and so through these examples, hopefully we're in a better place to be able to start computing derivatives of trigonometric functions that might involve the chain rule It's very common to compose trigonometric functions in this manner So the chain rule is a critical tool to help you in those examples