 So how about those trigonometric functions? So in general, we have the basic derivatives of the trigonometric functions. The derivative of sine is cosine, or again, in differential notation, the derivative of sine is cosine, and likewise, the derivative of cosine is minus sine of x, and the derivative of cosine again minus sine of x. These are the only two you actually need to remember, and actually, you only need this one if you remember all of your trigonometric identities. Ah, for it's convenient to have a couple of others. So from these and the basic trigonometric identities, we also find the derivatives of the other important trigonometric functions. The derivative of tangent is secant squared of x, and likewise, again, in prime notation, and the derivative of secant is secant tangent of x. Again, we'll express that in prime notation. The two important things to remember when working with trigonometric functions are the same as with working with any functions. Remember the chain rule. The chain rule is the most important component of almost any derivative. So for example, let's say I want to find the derivative of sine of 8x plus 8, and again, a little bit of analysis goes a long way. What do we do? If we want to evaluate this function, take x times 8 plus 8, hold that thought, then take the sine of whatever that value is. So the last thing we do is to take the sine of a number, and so this is a sine function. So that's the first thing we'll differentiate. And again, useful idea for applying the chain. We'll just drop everything except for that last function. This is the derivative of sine, and we remember that the derivative of sine is cosine times the derivative of whatever's inside the parentheses. And again, we apply the kindergarten rule. If you take something out, put it back. And so that 8x plus 8 should be in there. I have cosine. I need the derivative of 8x plus 8, which is just going to be 8. And one thing that's worth noting here is check for the echo of the original function in the derivative. So again, we have 8x plus 8 up here. We have 8x plus 8 in our derivative. And that says that we are probably heading in the right direction. Well, that was fine. Let's do another one. So here I have a function secant x squared plus 5, and I want to find the derivative of the function. So again, a little bit of analysis goes a long way. I'm going to take x. I'm going to square it. I'm going to add 5, figure out what that number is, and then take the secant of that value. And that gives me my function value. The last thing I do is find secant. So this is a secant function. So my function, drop everything except for the last thing that we do. And then I know how to differentiate secant. The derivative of secant is secant tangent times the derivative of something. Again, you can never go wrong by applying the chain rule too many times. And again, we'll apply the kindergarten rule. If you take something out, put it back in. Now, I don't need to do anything with this. Don't need to do anything with that. This last term says derivative of x squared plus 5. So I'll evaluate that. And there's my derivative. And again, check for the echo. We have an x squared plus 5 here. We have an x squared plus 5 here. And a secondary echo, x squared plus 5 here as well. How about a more complicated function? Cosine e to the power of 3x minus 5. And again, we'll go through the analysis. Take x times 3 minus 5. e to whatever that number is. Cosine of whatever that number is. The last thing we do is take the cosine. So this is a cosine function. So if I want to find the derivative, I'll drop everything except for that last function. This is the derivative of cosine. And derivative of cosine is minus sine, whatever. And again, kindergarten rule. If you take something out, put it back in. So that's minus sine e to the 3x minus 5. Don't need to do anything with this. This is complete. And then I need to find the derivative of e to power 3x minus 5. And again, a little bit of analysis goes a long way. What is this? This is take x times 3 subtract 5. e raised to that number. So what I'm differentiating is an e raised to the type of function. So I'll drop everything except for the last thing that we do. This is the derivative of e to the something. And derivative of e to the something. World's easiest derivative is the same thing times the derivative of our something. We apply the kindergarten rule. We took it out. We'll put it back in. And last step, derivative of 3x minus 5. That's just a straightforward function. And there's my derivative.