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Multivariable chain rule

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Presentation on theme: "Multivariable chain rule"— Presentation transcript:

1 Multivariable chain rule
Math 200 Week 5 - Monday Multivariable chain rule

2 Math 200 Goals Be able to compute partial derivatives with the various versions of the multivariate chain rule. Be able to compare your answer with the direct method of computing the partial derivatives. Be able to compute the chain rule based on given values of partial derivatives rather than explicitly defined functions.

3 Math 200 What we know From Calc 1 we know that to differentiate a composition of functions, we need the chain rule We can also write it like this: Where u=g(x) E.g. y = sin(3x2) y’ = 6xcos(3x2)

4 Math 200 Heres a more visual way of thinking about the chain rule: We have x being plugged into g and then g(x) being plugged into f. Then out comes f(g(x)) g f x g(x) f(g(x)) g’(x) f’(g(x)) So, when differentiating f(g(x)) with respect to x, you have to work your way back one “layer” at a time

5 Math 200 New stuff Suppose we have a function of two variables, z(x,y) Let x and y be defined in terms of two other variables, s and t x = x(s,t); y=y(s,t) We need a diagram like the one on the previous slide but with multiple paths to the result, z(x,y) z x y s t s t

6 Multivariable Chain Rule
Math 200 Say we want to differentiate z with respect to t… z x y s t We have to work our way from the top of the tree diagram to the t’s on the bottom layer along every possible path Multivariable Chain Rule

7 Example Let f(x,y) = x2 + y2 Also let x = sin(st) and y = s2t3 Compute
Math 200 Example Let f(x,y) = x2 + y2 Also let x = sin(st) and y = s2t3 Compute Start by getting all the pieces together…

8 Math 200 Wait… If I can just plug into x and y at the end, why can’t I just do that to begin with? Let’s try it Now, just differentiate with respect to t

9 Math 200 What does it mean? First, it’s worth noting that the multivariate chain rule is more of a conceptual tool than a computational tool E.g., we’re going to use it to define the directional derivative soon We can use it to check our work when we take the shortcut to plug in before differentiating We can take numerical information about functions and their partial derivatives and apply the chain rule

10 Example Consider the function z = xy, where x=est and y=s2 + t3
Math 200 Example Consider the function z = xy, where x=est and y=s2 + t3 Compute Use the chain rule for practice and then check your work by plugging in first and then differentiating

11

12 Math 200 Another Example Suppose f is a differentiable function of x and y and that Use the following table to evaluate

13 We’re asked to evaluate
Math 200 We’re asked to evaluate But the table uses (x,y) values not (s,t) values x(1,-1)=3 and y(1,-1)=-2 so rather than fx(1,-2) and fy(1,-2), we want fx(3,-2) and fy(3,-2) First, let’s just rewrite out the chain rule as it applies here: From the definition of z, we know that x=2s-t and y=s+3t

14 Math 200 One more example Suppose w = w(x,y,z) and that x=x(a,b,c), y=y(a,b,c), and z=z(a,b,c) i.e. w is a function of x,y, and z and x,y, and z are each functions of a, b, and c Write out the chain rule for

15 w z y x c b a c b a c b a The diagram looks like this
Math 200 The diagram looks like this w z y x c b a c b a c b a So we want to follow every possible branch from w to b


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