Special Functions Trigonometric Functions Taking the Derivative of Trig Functions Page [1 of 3]

All right. Well, now that we sort of have a vague notion about these trig functions, let's see if we can actually figure what the derivatives of these trigonometric functions are. Now in fact, if you really wanted to figure out for example, the derivative of the sine function, let's start with sine. What would you do? Well, if you want to rigorously discover and prove what the derivative of the sine function is what would you have to do? Well, needless to say what you'd have to do is you'd have to start back at the very beginning. Return to the derivative of the definition, and as the limit, as delta x goes to zero, f of x plus delta x minus f of x all over delta x. And then you would insert for the function f, sine of x, until you see sine of x plus delta x minus sine of x, all over delta x. You’d use the fact that sine of a sum of two angles, actually is the formula. It's the sum formula for the sine function. And then you'd have to do some simplification and look at some complicated limits, cause you'll get an indeterminate form and look at that limit and figure it out. Okay and then, you get the answer. Instead of going through that process, which I don't think is particularly that intuitive or interesting, I thought it is better to remember the fundamental fact that the derivative represents the slope of the tangent line to the function. So instead, let's do this pictorially and try to inspire and make an educated guess as to what we think the derivative of sine of x should be, and then we'll see that, in fact, that's what it really is. The way to do that again is to look at a picture. Cause you know, it's more fun to look at pictures, than to deal with a lot of formulas and stuff. So here again is the sine function. What I'd like for us to do is take a look at what the slopes of the tangent lines are here. So if I put the tangent line right over here, let's say, you'll see the tangent is positive which means the slope therefore, is positive. So the derivative is positive, so at this point the derivative should be positive. Okay, now what happens as we move along? Well, the derivative is still positive, but notice that it's decreasing, because in fact I'm leveling off. So there's still positive numbers but they're getting smaller and smaller. And then what happens here? Well, here the derivative is actually zero. The tangent line is horizontal at that point, the derivative is zero. What happens when I keep continuing? When I continue onward, you see that now the slopes begin to take on negative values because I'm dipping down. And so now I'm negative, I'm more negative. So I'm getting further and further down in the negative numbers and I keep getting down and then right around here, notice, I start to – now it's still negative, but I'm less negative. Right I'm moving toward zero, so I'm increasing. I'm increasing, I'm increasing, and then all of the sudden, whamo! I'm back level again, so the derivative there is zero. And then what happens after that point? Well, the tangents there now become, now become positive again, and they're positive. So what do I see? I see that the derivative is positive, then it becomes – it's positive. Then it becomes zero, then it becomes negative, then it starts to increase and become less negative. Then it becomes zero again and then it becomes positive. So either function that starts off positive and goes down to zero and keeps going down, but then comes up again and goes back to zero and then keeps going up. What kind of function does that? Well, let’s see if we can actually look at a graph of one. So let's see if we can look at a graph of one here, what would that look like? So I'm not going to plot the values of the slopes, plot the values of the slopes. Let's draw a little axis here. The values of the slopes -- so here I'm very positive, here I'm very positive. So I put that positive number up here. And then what happens, as I keep going, I'm still positive but I'm getting smaller and smaller and smaller, until I get to zero. So I'm still positive but I'm getting smaller and smaller and smaller, somehow, until I get to zero. Now I'm at zero. Then what happens? Well, when I keep getting negative, I get more and more negative, more and more and more negative. So I'm going to get more and more and more and more negative. And then what happens? Well, then what happens is, I actually uh, get more and more negative. But then I start getting less and less and less and less negative, and then I come back up to here, where the slope is zero again. So now I have to be at that point, so the slope has to be zero. Remember these are the graphs of the numbers of the slopes. And then what happens after that? Well, now I'm positive again. So the slope is positive, I should be going up and I'm increasing, so it's looks like that. So what kind of function has that basic shape of starting up and coming down, coming way down, and then coming back up again, and then repeating. Well, if you think about that, that turns out to be, at least in spirit, the cosine function. The cosine function starts up here, then comes down and then goes up. And in fact, if you put that right underneath there, you can really see that business.

Special Functions Trigonometric Functions Taking the Derivative of Trig Functions Page [2 of 3]

In fact, let me do this for you right now, live. What I'm going to do is, I'm just going to start here, and I'm going to be calling off the values of the slope numbers. And you see if they correspond with the values of this function, the way this function is plotted. So here it’s very positive and in fact, it turns out to be one, so there's one. Now what happened? I'm still positive, but I'm decreasing, still positive, but decreasing, still positive, decreasing, decreasing, decreasing, then I get to zero, right here. And notice right there, I'm at zero. Now what happens? Well, now I'm decreasing. I'm negative; that should be negative, I'm negative, I'm negative. I'm decreasing, decreasing, decreasing, until I get to here. Now I'm really negative, slope there is negative one. And now I'm still negative, but I begin to start going on an up, increasing, I'm still negative, increasing, still negative but going up now till I get to here. That's zero and that's at zero. Now what happens – well, now I'm positive, ping. I'm positive and what happens. I keep increasing, increasing, increasing, increasing, increasing, until I get to here. So in fact, the derivative of sine, we can now guess, is cosine, and that guess turns out to be correct and you can verify that going back to the definition of the derivative. Let me write that down for you. What we see is, is f of x equals the sine of x, then f prime of x equals the cosine of x. So there's the derivative of sine, which we hopefully get an intuitive feel for, just by looking at the graphs and remembering that the derivative represents the slope and seeing how the slopes vary. Okay, now what about, what about the cosine functions. Let's look at f of x equals cosine of x. How would that work? Well, let's look at cosine. You might in fact, guess right out of the get go that the derivative might be sine. So in fact, let's just draw a little picture of that to see if that's a good guess or not. Cause if the derivative of sine was cosine, it's reasonable to guess that the derivative of cosine is sine. I think that's a great guess, by the way. Did you make that guess? I did! So there's the sine function, it keeps going of course. And let's see if that's a good guess. Cause, remember if that's the derivative, that should emulate the activity of the slope of the tangent line, tangent to my face, whoop, okay. As you start up here, we can see that the slope of the tangent line is zero, because that tangent is horizontal, and that's good because look at the value there is zero. But what happens as I move? As I move, I see my values are actually now sloped negatively. So in fact, these should be negative and that's a problem because this is now going up, it's positive. These should be negative, negative, negative, for a while, and then be less and less and less negative until we get back to zero. This should actually dip below, dip below, and then come back up to zero. And then what happens here? Well, then I start to go up, up, up, up, it's still negative. I'm sorry, I'm down at zero, but I go up, up, up, on positive, so now I should be going up, up, up, positive until I get to here. And then I start to level off again, so I should be going up, up, up, up, up, and then down, until I get to here back at zero. So this is far from the answer, and so what I wanted to do is I wanted the thing to drop first, and then I want it to be zero, and then I want it to be zero right here. And then I want it to start to go up again until I'm back to zero here. Well, that's not the sine, but notice, if I flip the picture, it actually tells you what I want. Because notice here, I start to drop, the slopes there are negative and these values are negatively valued. And then right here, my slope – it turns out to actually be negative one, which is this point right here. And then afterwards, they still are negative, but they're increasing more, less and less negative, until they get to here, and here the slope is zero, horizontal, and that's zero. And then the slopes become positive again. Great, I'm going up. Right, positive values, positive values, positive values, and then I come down again and over here, I land in zero. So this seems to be the right curve. This was the sine curve. What do I get if I just flip all the values? I take all the positive values, and make them negative, take all the negative values and make them positive. I'm basically just taking the negative of this function – if I take the negative of this function, it flips it. So in fact, we see that a good guess for the derivative of cosine is negative sine. Cause I got to go down first, and then up, and it turns out that's actually the right answer. So the derivative of cosine is not sine, but is instead, negative sine. And you can see that, by thinking about how the slopes are changing and you can really see that, that picture. So in fact, we've just discovered that the derivative of cosine, a little bit more surprising than you may have thought, you see, it's negative sine of x. That's a little surprise there, the trig functions are always, always there, filled with surprises.

Special Functions Trigonometric Functions Taking the Derivative of Trig Functions Page [3 of 3]

Okay, now what about the tangent? Well, now we're actually armed to display the tangent function. Because remember once you know everything about the sine and cosine, you are done. Let's look at the tangent together, f of x equals the tangent of x. And I want to take the derivative of x, so what do I do? Well, I don't know what the derivative is, that's what I'm trying to figure out. I do know that this equals sine of x over cosine of x. That was one of the first identities that we discovered in the previous review section. So I can actually use the Quotient Rule here – right, remember the Quotient Rule, the rule chant. Bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared. So let's actually perform that right now. And what do I see? I see the bottom. That's cosine of x multiplied by the derivative of the top, and we just saw the derivative of sine is, you can see it right there, cosine. And now where am I in the Quotient Rule? We can actually start over and practice. The bottom times the derivative of the top – okay, minus. There's my chant, the top times the derivative of the bottom. What’s the derivative of cosine? We just saw it’s negative sine, and it's all divided by the bottom squared, so cosine squared actually. Remember that's how I write cosine of x, all squared. I put the little two there, that means cosine x multiplied by cosine x, cosine x squared. Okay, let's simplify this a little bit. Cosine x times cosine x – well, that's cosine x squared. The cosine squared x and a minus and a minus make a plus, and then I have sine times sine. That's sine squared x, all over cosine squared x. Well, that looks pretty complicated, but I guess that's the answer. Could we simplify that at all? Well, yeah, because remember the fundamental fact that sine squared x plus cosine squared x always equals the constant number one, no matter what x is. This we saw -- this is all from the Pythagorean theorem, if you remember in the review section we just had. This is just the number one, so in fact I can simplify this dramatically, just one over cosine squared x and what is one over cosine? You may remember that's actually secant. So instead of one over cosine squared, I could write this as secant squared x. We just proved that the derivative of tangent is actually secant squared x. We can come back here and report in this little list we have here, that the derivative of tangent -- just using the Quotient Rule and these two facts, you don't have to memorize anything -- is equal to secant squared x. What about the other trig functions, by the way? Like what about, what about cosecant. Well, if you wanted to find the derivative of cosecant, what would you do? Well, you'd remember that cosecant is equal to one over, one over sine, so if you wanted cosecant x, you remember that equals one over sine, and now you can use the Quotient Rule. Bottom times derivative of the top minus the top times the derivative of the bottom squared you can see exactly what it equals. Why don’t you try those other functions, those ones that are not the stars but just the supporting cast? You can now find the derivatives of any, of any function, of any trig functions, rather. So now, we can take derivative of trig functions, terrific. How about exponential functions, how about logarithm functions? Well, everyone hates logs and exponentials. So up next, we'll first actually do a review of the very important and natural functions of logs and exponentials, which actually capture the spirit of growth and growth rate. And then, we'll take a look at the derivatives of those things. Okay, I'll see you over in a while. Bye.