1. Math 200
Week 4 - Monday
Partial derivatives

2. Math 200
Goals
Figure out how to take derivatives of functions of multiple variables and what those derivatives mean.
Be able to compute first-order and second-order partial derivatives.
Be able to perform implicit partial differentiation.
Be able to solve various word problems involving rates of change, which use partial derivatives.

3. Review The point (1,0) is on the graph of f
Math 200
Review
Calculus I:
dy/dx = f’(x)
f’(a) = “slope of the tangent line to f at x = a”
e.g.
The point (1,0) is on the graph of f
The slope of the line tangent to f at (1,0) is 2

4. Consider a 3d example Let f(x,y) = x2 + y2
Math 200
Consider a 3d example
Let f(x,y) = x2 + y2
Consider the trace of f on the plane y=1
f(x,1) = x2 + 1

5. Differentiating with respect to x, we get dz/dx = 2x
Math 200
We can certainly find slope of the line tangent to z = x2+1 at any point on the xz-plane…
Differentiating with respect to x, we get dz/dx = 2x
The slope of the line at x=1 is 2

6. The slope of 2 is telling us [change in z]/[change in x]
Math 200
So, can we write parametric equations or a vector-valued function for that same line in 3-Space…?
Need: (1) a point on the line and (2) a direction vector parallel to the line
We were at x=1 on the plane y=1. Since f(1,1) = 2, the point of tangency is (1,1,2)
The slope of 2 is telling us [change in z]/[change in x]
We need a direction vector for which z/x = 2 and y=0…
<1,0,2> works! (There are infinitely many other choices of course.

7. We’d be looking at the trace f(x,2) = x2 + 4
Math 200
What if we repeat the same process for the trace of f on the plane y=2?
We’d be looking at the trace f(x,2) = x2 + 4
Here it is on the xz-plane with its tangent line at x=1

8. Differentiating with respect to x, we get dz/dx = 2x It’s the same!
Math 200
We can find slope of the line tangent to z = x2+4 at any point on the xz-plane…
Differentiating with respect to x, we get dz/dx = 2x
It’s the same!
The slope of the line at x=1 is 2
Since [change in z]/[change in x] = 2, we can use the direction vector <1,0,2> again, with starting point (1,2,5)

9. What just happened…? We just our first partial derivative!
Math 200
What just happened…?
We just our first partial derivative!
Notice that in both cases (whether we set y=1 or y=2) we got dz/dx = 2x
This would’ve been the case with any choice of constant value for y
We could have done the same work on any plane of the form x=constant
In that case, we’d find [change in z]/[change in y]

10. Math 200
Definitions
The partial derivative of f with respect to x is what you get when you…
treat y as a constant…
and differentiate with respect to x
We write any of the following:
We use this partial symbol instead of just d to indicate that there is more than one independent variable

11. Definitions Notice: We’re not using “prime” notation anymore…
Math 200
Definitions
The partial derivative of f with respect to y is what you get when you…
treat x as a constant…
and differentiate with respect to y
We write any of the following:
Notice: We’re not using “prime” notation anymore…
If I write f’(x,y), you don’t know which variable I’m holding constant

12. Math 200
A little practice
Compute both first-order partial derivatives (fancy way of saying first derivatives) for the following functions
i.e. compute the partial derivative with respect to x and the partial derivative with respect to y for each function

13. Example 1 with respect to x, These are constant terms Power rule
Math 200
with respect to x, These are constant terms
Power rule
Example 1
with respect to y, These are constant terms
Power rule

14. with respect to x, this Whole term is constant
Math 200
with respect to x, this Whole term is constant
Example 2
y2 is treated as constant here, so it’s like differentiating 5x with respect to x
x is treated as constant here, so it’s like differentiating 5y2 with respect to y

15. Math 200
the derivative of eu(x) is u’(x)eu(x), so in terms of partial derivatives, we should write ux(x,y)eu(x,y)
Example 3
the derivative of eu(x) is u’(x)eu(x), so in terms of partial derivatives, we should write uy(x,y)eu(x,y)

16. Math 200
Example 4
We can write “The partial derivative with respect to x of x2y3” like this:

17. What partial derivatives give us
Math 200
What partial derivatives give us
Let’s look at f(x,y) = 3x2 - 2y + 1 from Example 1 at the point (1,2)
We found that fx(x,y) = 6x
Evaluating the x partial at (1,2) we get fx(1,2) = 6(1) = 6
What does this 6 tell us?
The rate of change of f (or z) in the x-direction at (1,2) is 6
The slope of the line tangent to the trace of f(x,y) on the plane y=2 is 6

18. Math 200

19. Higher order partial derivatives
Math 200
Higher order partial derivatives
Consider the function z = 3x2 - x3y4
Let’s find the two first order partial derivatives: —
We could now differentiate either of these with respect to x or with respect to y…
…for a total of four second order partial derivatives

20. Notice that the second order mixed partials are the same!
Math 200
Notice that the second order mixed partials are the same!
This will be true for any time they are both continuous — — —

21. Implicit differentiation
Math 200
Implicit differentiation
Recall from calc 1:
Treat y as an implicit function of x

22. Implicit Differentiation with three variables
Math 200
Implicit Differentiation with three variables
Treat z as a function of x and y as a constant

23. Get all of the dy/dz terms on one side and factor
Math 200
One more example
Compute dy/dz for
Differentiate both sides with respect to z, treating y as a function of z and x as a constant
Product rule
Get all of the dy/dz terms on one side and factor
Solve

24. Math 200
Recap
Trace on y=y0
To compute the partial derivative of f with respect to x, we…
Treat all other variables as constants
Use all of the derivative rules we know from Calculus 1
E.g. f(x,y) = x3+y3
fx(x,y) = 3x2
When we evaluate partial derivative of f with respect to x at a point (x0,y0), we get the slope of the line tangent to the trace of f on the plane y = y0
(x0,y0)