[MUSIC] In today's video, we introduce, explain, and illustrate the Quotient Rule which enables us to differentiate a quotient of expressions in terms of the derivatives of the numerator and denominator. Consider a function y of x which is a quotient u divided by v where u and v are themselves functions of x. The quotient rule states that the derivative of y with respect to x is v times the derivative of u, minus u times the derivative of v all over v squared, expressed this way using Leibniz notation. Equivalently, using function notation and a dash to denote the derivative, the quotient rule says y dashed is vu dashed minus uv dashed all over v squared. It might look complicated at first, but it's remarkably elegant considering the task at hand, which is to differentiate a possibly highly sophisticated function expressed as a quotient. That we can do it at all is surely something of a miracle. The quotient rule, in fact, follows from the product and the chain rules. The reason is worth investigating, as it consolidates your understanding of these rules and the way they fit together and makes your technique more robust and stronger. Put y equal to u over v, which we can write as a product u times 1 over v. Which we can further rewrite as u times v to the minus 1. Because this expression is a product, naturally we can apply the product rule. So that dy/dx equals the first factor u times the derivative of the second factor. But the second factor v to the minus 1 times the derivative of the first factor. But the derivative of v to the minus 1 with respect to x may be expanded to become d/dv of v to the minus 1 times dv/dx by the chain rule, which becomes minus 1 times v to the minus 2 times dv/dx. Which can be rewritten as negative dx dived by v squared. Hence the dy/dx becomes u times this expression plus the second piece which can be rewritten as dudx/v. And the first and second pieces can be rewritten so the whole thing can becomes v du/dx minus u du/dx all over v squared. Which is exactly the form of the derivative in the quotient rule. This produces the Leibniz form which immediately converts into the rule using dash notation. For example, let y equal quotient x squared plus 1 divided by x. And our task is to find the derivative, y dash. We can solve this directly by splitting y up into two fractions, which becomes x plus x to the minus 1, and then differentiate each piece. The derivative of the first piece is 1 and the derivative of the second piece is minus 1 times x to the minus 2. Which becomes 1 minus 1 over x squared, which we can leave as that or rewrite as a single fraction, x squared minus 1 over x squared. An alternative solution is to apply the quotient rule because y is the quotient of u divided by v where u is the numerator x squared plus 1, and v is the denominator x. So that u dashed is 2x, and v dashed is 1. So by the quotient rule, y dashed is vu dash minus uv dash all over v squared, and we can put all the pieces together. Expand, simplifying a couple steps to get x squared minus 1 over x squared which agrees with our first direct solution. Here's a more complicated expression for y and our task again is to find the derivative y dash. Observe that y is the quotient, u divided by v, where u is the numerator and v is the denominator with derivatives u dash and v dash respectively. We apply the quotient rule, and put all the pieces together. The derivative of y dash is complicated, and I'm not even going to try to simplify it. I just made up this example for the purposes of illustration, and have no idea what the graph of the function looks like, or even whether the function corresponds to some interesting or important problem, or physical phenomenon. The point I want to emphasize in an example like this is that we can find the derivative quickly by means of the rules that we developed. And the process becomes quite mechanical and procedural. Something quite remarkable has happened. Just in the space of a few videos, we have moved from basic definitions of the derivative to being able to differentiate such sophisticated functions fluently and easily. Even though we don't know what the curve looks like, this expression for the derivative, in principle, allows us to find slopes of tangent lines at any given point in which the expression is defined. And therefore, to find the equation or the tangent line at a given point and thereby approximate values of the function. In my opinion, this is an extraordinary achievement. And if you follow the development up to this point, then you should be congratulated on coming so far so quickly. In an earlier video I told you what the derivative of tan x turns out to be. Let's see if we can discover it ourselves. Observe that tan x is sin x over cos x, which becomes a fraction u over v where u equals sin x and v equals cos x. So that u dash is cos x, And v dashed is minus sin x. By the quotient rule, the derivative of tan x with respect to x becomes vu dash minus uv dash all over v squared. And we put the pieces together. And the denominator becomes cos squared x. Whilst, the numerator becomes cos squared x plus sin squared x, which you might recognize as simplifying to 1 by the circular identity. And the answer becomes 1 over cos squared x. This agrees with the answer that I told you some time ago without giving any reason. Now, there's some common notation and terminology that you should become aware of involving another trig function called the secant. We put sec x equal to 1 over cos x, the reciprocal of cos x, where sec is an abbreviation for secant. This word is related to lines, drawing certain points in diagrams involving the unit circle, in the context of more advanced trigonometry, but I won't go into the details. And you can look it up yourselves, if you wish, out of interest. This new terminology, the derivative of tan x becomes a square of sec x, which is written as sec squared x. You can continue to think of the derivative as 1 over cos squared x if you like. But you'll almost certainly see the sec notation in your mathematical travels. The secant function simply reciprocates the value of the cosine function. There's special terminology also for reciprocating the value of the sine function. Put csc x equal to cosec x equal to 1 over sine x, where csc and cosec are both common abbreviations for cosecant. There's one more, put cot x equal to cotan x equal cos x divided by sin x, which is the reciprocal of tan x. And here cot and cotan are common abbreviations for cotangent. Thus cosecant and cotangent, again have geometric interpretations in more advanced trigonometry which I won't go into here. And we will not need either of them in this course. We have from before, the derivative tan x is 1 over cos squared x expressed also as sec squared x. For completeness, though we will not need it, and you can check it yourself if you wish, the derivative cot x or cotan x is negative 1 over sin squared x. Which you can express as negative cos x squared x using either abbreviation. In today's video, we introduced and illustrated the quotient rule. Which enables you to differentiate a quotient of expressions in terms of the derivatives of the numerator and denominator. We explained how it comes about as results of applying the product and chain rules. By expressing the quotient as a product, the numerator with the reciprocal of the denominator. Ee gave some contrasting examples and importantly, found a formula for the derivative of tan x. Namely, the reciprocal of cos squared x, which can also be expressed as sec squared x, where sec is an abbreviation for secant. The values of the secant function are the reciprocals of the values of the cosine function. Please read the notes, and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon. [MUSIC]