 So what about the derivative of sine? Well, let's see what it can say about it first before we find it. If you only learn one thing from calculus, you'll probably have failed the course. But among the things you should learn is that the derivative is the slope of the tangent line. So let's consider the graph of y equals sine of x. Now, since the function is periodic, the slopes of the tangent lines are also periodic, which means that the derivative will also be periodic with the same period. Now, if we look a little closer, we see that the derivative of sine must begin with some positive value, drop down to 0 at x equals pi halves, become negative, and then return to 0 at x equals 3 pi halves, and then become positive again at x equals 2 pi. And this suggests the derivative of sine x is something like cosine x. So we should expect to find the derivative of sine to be something cosine x. Well, let's find it. So by definition, the derivative is the limit of the difference quotient. Now, here we have the sine of a sum, and so we can use the angle sum identity to rewrite it. And we have this horrifying mess, and let's see if we can rewrite this. Now, our goal when we rewrite this is we know the limit as h approaches 0 of sine h divided by h. So let's try and get as many sine h divided by h as we could possibly get. So here's one, so let's split that one off, and in fact, let's isolate that sine h divided by h. Now, we also have some terms left over. In this other fraction, there's a common factor of sine x we can remove, and this is a cosine h, and we don't know what to do with cosine h, but there is a connection between sine and cosine, and we can get there if we multiply, numerator and denominator by cosine h plus 1, because if we do that, we get, and this cosine squared h minus 1, we can use the Pythagorean identity and make that minus sine squared h. And again, since our denominators are being multiplied, we can change the order, and again we want to split off a sine h divided by h, and that leaves us with leftovers minus sine h and all the rest of it. And that means we can rewrite this horrifying mess. And now we need to find this limit. Well, it's the limit of a sum, and so that's going to be the sum of the limits, and this first limit, well, that's a product, and so the limit of a product is the product of the limits, as h goes to 0, cosine x doesn't do anything, and sine h divided by h goes to 1, and so this limit will just be cosine x. Now, this other limit, it's a product, and so the limit of a product is the product of the limits, so we'll split that into the three factors. In our first factor, as h goes to 0, this expression goes to... In our second expression, as h goes to 0, this goes to... And since we're multiplying these together, it really doesn't matter what the other limit does, but it's worth remembering that the limit as h goes to 0 of sine h over h is equal to 1, and the limit of the product is the product of these three, 0, and so our derivative of sine is equal to cosine. And the important thing to note here is that this is consistent with our expectations that the derivative of sine is going to be cosine something. Now, once we have the derivative of sine equal to cosine, we can find the derivatives of all the other trigonometric functions from some combination of the chain rule or implicit differentiation. In fact, we can find the derivative of cosine in several different ways. First, since cosine is sine pi halves minus x, then the derivative of cosine is the same as the derivative of sine pi halves minus x. And we can then use the chain rule to find and simplify our derivative. Or maybe that's too easy. We can use our Pythagorean identity sine squared plus cosine squared equals 1. If we differentiate it, we can use implicit differentiation and solve for the derivative of cosine. Or maybe we want to be really creative since we know other relations between sine and cosine, we can use any of them to find the derivative of cosine x. And all we have to do is apply the chain rule, implicit differentiation, and some knowledge of the trigonometric functions and various identities. And in this way, we can find the derivative of cosine as well as the derivatives of the other trigonometric functions.