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

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