when considering families of functions and their derivatives an important family for scientific applications is of course the trig family sine cosine tangent and all the related functions to actually derive these rules though is surprisingly complicated when you go back the definition and requires working carefully with limits and trig identities in practice that's interesting and from a mathematical perspective but it's often not very helpful from an intuitive sense so what we're going to do is derive the formulas for the derivative of sine and cosine anyway in a graphical manner and then extend that using our derivative rules like the chain rule to get all the other rules that we need here is the graph of sine of x goes up to one goes down to negative one has its zeros at zero pi two pi and so on now remembering what we mean by the relationship between the graph of f of x and its derivative if we take a look at this graph and look at slopes we're going to see values on this function here what do i mean by that we take a point we see a slope of zero that means we're going to get a value a dot on this graph at height zero likewise at this point here the slope is zero we're going to get a dot at value zero and same thing here peak and valley slope of zero means a value f prime of zero slope of zero f prime of zero what can we say in between those here we can say that f of x is increasing the corresponding interpretation or the way we visualize that on the derivative graph well if our function is increasing our derivative is positive and we can get even a little more specific with that this graph is actually to scale and 1.7 here is a bit longer than one a bit further than one 1.6 ish and the slope here is actually exactly one this is another reason why we use radians in calculus class when we talk about trig functions if we scale the axes here so one cycle takes 2 pi then this slope is exactly 1 and then we see the slopes get lower and lower and lower and hit zero so the values of the derivative get smaller and smaller until they hit zero and likewise here the values are zero then they get bigger and bigger and bigger zero bigger and bigger and [Music] oh bigger like to see a nice arch like this on the next interval we see that our function is decreasing and the translation about the derivative then we say the derivative is negative on this next interval well not too surprisingly in the middle here we expect to see a slope of the same as this but negative so we see a slope of negative one and hey keep repeating those patterns and pretty soon you start seeing a graph that looks an awful lot like starts at its peak goes down has an amplitude of 1 period of 2 pi we have the derivative of sine of x is cos of x this is not a proof by any stretch of the imagination but it ties in graphically or builds a relationship graphically between those two functions so it's certainly plausible that the derivative of sine of x is cos of x and that is in fact true let's immortalize that in a theorem box just to get that down for posterity and then we'll take a look at the derivatives of all the other trig functions that are commonly used here is this table of other derivatives the one gotcha one that is most commonly used is the derivative of cos is negative sign so just be aware that that negative creeps in the derivative of tan is secant squared you can actually derive this and all the other rules from basic trig identities and the derivative rules if you take the derivative of tan that's the same as the derivative of sine of x over cos of x and if you do some work you can get back to secant of x just remembering that secant of x is one over cos so secant squared of x is one over cos squared so this rule if you don't remember it off the top of your head you can derive it in about three steps using the quotient rule and the definition of tan the same is true for these other three derivatives it's up to you how you want to remember them the one pattern that does stick with me is that if there's a c in the function itself cosecant cos or cotangent those derivatives all have negatives in them sine tan and secant all have positive derivatives there's some other structure about secant squared versus cosecant squared that kind of thing whatever mnemonic works for you the main ones of course though are the derivative of sines cos and the derivative of cos is negative sine let's do a quick proof of the secant rule just to see how that plays out the derivative of the secant of x is more easily stated as the derivative of 1 over cos of x because we know the rule for cos of x and that is better stated as the derivative of cos of x all to the power negative one that'll be another way to write the same thing and now we have a classic chain rule application where we have the derivative of something to the negative one and then we get the same thing inside cos of x to the negative two and then we take the derivative of the inside and the derivative of cosine is negative sine of x that doesn't get us all the way to the end but we can see some cancellations we have a negative and a negative so we're going to get a positive overall cosine to the negative 2 is 1 over cosine squared of x and then we have a sine of x left over that's a perfectly acceptable form of the derivative historically though we tend to put it in this form here and it's not too hard to do all we do is take this cosine of x all squared and break it into one cos of x in one term times the sine of x with the other cos of x so we split this cos squared into one cos on the left one cos on the right and lo and behold sine over cos is what we define as tan and one over cos is where we started all this with secant so the derivative of secant x is secant x times tan of x that's a quick practice challenge take a look at this function here and see using the derivative rules for trig functions now and the chain rule can you identify the derivative here in the list i'll pause for a second then we'll come back all right the derivative of four plus six cosine of pi x squared plus one is going to be derivative of a constant by itself is zero the six is a multiplier so it stays around the derivative of cos anything from the chain rule is going to be negative sine of the same thing and then we take the derivative of the inside and multiply by that so we're going to have pi times 2x all of that together should look like one of the expressions we have here we've got negative 6 sine times 2 pi x that leaves us with c being the correct answer we could tidy up a bit more if we wanted to but here is just enough to get the derivative value as a function