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Chain Rule.Partial Differentiation.Young’s Theorem

Chain Rule.Partial Differentiation.Young’s Theorem.The idea of a derivative.How much is the value of y going to change when the value of x changes by a knownamount?Rates of change and amount of change.What is the amount of the change in y when ? What is the amount of thechange in y when ?Are these the same? When will they be the same? Think of a striaght line -are the rates of change the same or different along different part if thestraight line?Representing the derivativeFind the first derivative of with respect to .The first derivative is represented as(represented also as or or ): i.e. it is only a label for thederivative.Formally, you will see in mathematics textbooks, that the slope isstated something like:A local linear approximation is being aimed at here which in thelimit equals the slope.Does this matter for us in finance not usually and on this course,not at all.So, the slope of at is found by substituting intoEvaluating the derivativeEvaluating the derivative at ONE point.and so, for the function , we find that whenevaluated at , the slope has the value of .Evalauting at a DIFFERENT point.Similarly, at the slope of is 0.Another example.Lets take another expression. Say, .What is the expression for the derrivative (i.e. the slope)? Rememberthis is denoted as ?What is the value of the slope at x = 1?What is the value of the slope at x = 0?Classes Next WeekWe will discuss in the classes, next week, the concept you need to have aboutrates of change, which is here synonymous with the word derivaive and what thismeans for us on this course.Derivative Securities and Derivative of functionsRemember, though that the word derivative if typically used to refer to aspecific type of security that is traded (bought and sold) on financialmarkets to help buy and sell risk. So, watch out for the two differentmeaning and uses of the SAME word in the same discipline.A function with two variables and its slopesGraph of ; with x-y planes (blue) drawn at z = 2, z = 16 and at z = 30. Thecurves (green) show how the function z looks at those values of z. That is, with z fixedat a specific value, we see that different combinations of x and y can give the samevalue for z. You can point to the graph with the mouse (it will turn to a curved arrow)and move the curved arrow around to see the graph from different perspectives tounderstand what I mean. As a good exercise in mouse control, try to get one of theaxes to disappear, say try the y axis.Another function with two variables and its slopesGraph of with x-z planes (blue) drawn at y = 5, y = 50 and at y = 90.The curves (green) show how the function z looks like (when we resitrct the values of yas just noted). With the value of y fixed at 5, we see that as x changes, z does change.This is the idea behind the partial derivative.>The idea behind partial derivativesIf then how do we decide how much the rate of change in zGiven , say for example what is the effect on z of a change in y?What does this mean? Simple Idea difficult to write downThe answer can be found be computing the partial derivative of z w.r.t. y (i.e.with xfixed at 475, the middle green curve graphs the change in z as y changes).It could have been worse. Be happy!Formally this would be defined as:Does this foraml defintion matter for us in finance not usually and on thiscourse, not at all. It may begin to matter once you reach PhD level infinance. For now be happy!Chain RuleWhile a graph gives us way to fix these concepts, we need a more robust way tocompute the quantities that are of interest, especially when we cannot easily graphthese functions. Such as, when we have four or more variables. Lets use an intuitiveexample to further strengthen these basic ideas.Suppose we know that something that we are interested in, say the class of honours, wehope to get on this course, depends on a number of factors. Call the class of honours,C. The letter C then stands for, 1st class, 2.1, 2.2, etc.The class of honours will depend onSeveral FactorsEffort (e)how much effort you put in (effort, lets denote it with the letter ‘e’) andDifficulty (d)on how difficult the course is (difficulty, lets denote it with the letter’d’).Teaching Quality (t)It is also the case that both the effort you are motivated to putin and the difficulty of the course is influenced by how well itis taught (teaching, lets denote it with ‘d’)Specification of Chain Effects and factorsThe same thing we noted above using words can be state asThis now allows us to measure the impact of teaching quality on grade/marks asAn Numerical ExamplePlease see if you can get to yourself. (a simple test of how goodyour ability to manipulate linear algebra from school days)Generic Chain RuleSuppose we know that In other words, Z depends onseveral factors ( say, n factors, where n stands for 10 or 100, or whatever thenumber of factors are).Dependency of on m other components.Let say we find that each of the factors depend on several othercomponents (say, there are m components)….The total derivative (w.r.t. a particular ) now depends on each of thefactorsSo, we now get the following:Compact NotationAnother ExamplePlease see if you can get to yourself (a good test of where youstand on using simple derivative calulus and linear algebra).What Order in the Partial Derivatives?Best to see this through and example.ExampleSuppose we have a function of two variables:Partialsand thenand thenThe patternSo, it seems the order of differentiation does NOT matter in partial derivatives.Young’s TheoremIs a proof that this is the case. In finance we are not usually concerned withproofs. We are almost soley concerned with applications of such proofs andmore importantly being careful to remember where these proofs do NOT apply.Other Rules borrowed from MathematicsProduct RuleDivision RuleSuggestion:Convert the division rule to product rule easier to remember and apply(well in my opinion)

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