The document is notes for a lesson on partial derivatives. It introduces partial derivatives and their motivation as slopes of curves through a point on a multi-variable function. It defines partial derivatives mathematically and gives an example. It also discusses second partial derivatives and notes that mixed partials are always equal due to Clairaut's Theorem when the function is continuous. Finally, it provides an example of calculating second partial derivatives.

1. Lesson 19 (Section 15.3 and 15.5)
Partial Derivatives
Math 20
November 2, 2007
Announcements
Problem Set 7 on the website. Due November 7.
OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)

2. Outline
Partial Derivatives
Motivation
Definition
Other Notations
Worksheet
Second derivatives
Don’t worry about the mixed partials
Marginal Quantities

3. Motivation
Think back to your one-variable class. What do we use the
derivative for?

4. Motivation
Think back to your one-variable class. What do we use the
derivative for?

5. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point

6. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point
Instantaneous rate of change at a point

7. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point
Instantaneous rate of change at a point
Best linear approximation near a point . . .

8. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point
Instantaneous rate of change at a point
Best linear approximation near a point . . .
What is the analogue of tangent line for a function of two (or
more) variables?

9. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point
Instantaneous rate of change at a point
Best linear approximation near a point . . .
What is the analogue of tangent line for a function of two (or
more) variables? It’s a plane in R3 (or hyperplane in Rn+1 ).

10. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point
Instantaneous rate of change at a point
Best linear approximation near a point . . .
What is the analogue of tangent line for a function of two (or
more) variables? It’s a plane in R3 (or hyperplane in Rn+1 ). What
is its “slope”?

11. Motivation
Think back to your one-variable class. What do we use the
derivative for?
Slope of the tangent line at a point
Instantaneous rate of change at a point
Best linear approximation near a point . . .
What is the analogue of tangent line for a function of two (or
more) variables? It’s a plane in R3 (or hyperplane in Rn+1 ). What
is its “slope”?
Clearly, no single scalar can describe the slope. Just by looking at
the traces which make the graph, however, we can see some
“special” curves whose slopes might be significant.

12. Example
Let f = f (x, y ) = 4 − x 2 − 2y 4 . Look at the point P = (1, 1, 1)
on the graph of f .
0
-10
z
-20
-30
-2 -2
-1 -1
0 0
y x
1 1

13. There are two interesting curves going through the point P:
x → (x, 1, f (x, 1)) y → (1, y , f (1, y ))
Each of these is a one-variable function, so makes a curve, and has
a slope!

14. There are two interesting curves going through the point P:
x → (x, 1, f (x, 1)) y → (1, y , f (1, y ))
Each of these is a one-variable function, so makes a curve, and has
a slope!
d d
2 − x2 = −2x|x=1 = −2.
f (x, 1) =
dx dx
x=1 x−1
d d
3 − 2y 4 = −8y 3 = −8.
f (1, y ) = y =1
dy dx
y =1 y =1
We see that the tangent plane is spanned by these two
vectors/slopes.

16. Upshot
At a point on the graph of a function of several variables, there is
more than one “slope” because there is more than one “curve”
through the point. We can take a curve in each direction by
(temporarily) fixing all the variables except one and treating that
like a one-variable function. The derivative of that function is just
one of the partial derivatives of f .

17. Definition
∂f ∂f
Let f : Rn → R. We define the partial derivatives ∂x1 , ∂x2 , ...
∂f
∂xn at a point (a1 , a2 , . . . , an ) as
f (a1 + h, a2 , . . . , an ) − f (a1 , a2 , . . . , an )
∂f
(a1 , a2 , . . . , an ) = lim ,
∂x1 h
h→0
f (a1 , a2 + h, . . . , an ) − f (a1 , a2 , . . . , an )
∂f
(a1 , a2 , . . . , an ) = lim ,
∂x2 h
h→0
...
f (a1 , a2 , . . . , an + h) − f (a1 , a2 , . . . , an )
∂f
(a1 , a2 , . . . , an ) = lim
∂xn h
h→0

18. Example
Let f (x, y ) = x 3 − 3xy 2 . Find its partial derivatives.

19. Example
Let f (x, y ) = x 3 − 3xy 2 . Find its partial derivatives.
Solution
∂f
When finding ∂x , we hold y constant. So
∂f ∂
= 3x 2 − (3y 2 ) (x) = 3x 2 − 3y 2
∂x ∂x

20. Example
Let f (x, y ) = x 3 − 3xy 2 . Find its partial derivatives.
Solution
∂f
When finding ∂x , we hold y constant. So
∂f ∂
= 3x 2 − (3y 2 ) (x) = 3x 2 − 3y 2
∂x ∂x
Similarly,
∂f
= 0 − 3x(2y ) = −6xy
∂y

21. Other Notations
∂f ∂z ∂f (x, y )
= =zx =zx =fx (x, y ) =f1 (x, y ) =
∂x ∂x ∂x
∂f ∂z ∂f (x, y )
= =zy =zy =fy (x, y ) =f2 (x, y ) =
∂y ∂y ∂y

22. Other Notations
∂f ∂z ∂f (x, y )
= =zx =zx =fx (x, y ) =f1 (x, y ) =
∂x ∂x ∂x
∂f ∂z ∂f (x, y )
= =zy =zy =fy (x, y ) =f2 (x, y ) =
∂y ∂y ∂y
S&H prefer the numerical subscripts rather than variable names.
Other authors have different preferences.

23. Worksheet Problems 1–4

26. Outline
Partial Derivatives
Motivation
Definition
Other Notations
Worksheet
Second derivatives
Don’t worry about the mixed partials
Marginal Quantities

27. Second derivatives
If f (x, y ) is a function of two variables, each of its partial
derivatives are function of two variables, and we can hope that
they are differentiable, too. So we define the second partial
derivatives.
∂2f ∂ ∂f
= = fxx = f11
∂x 2 ∂x ∂x
∂2f ∂ ∂f
= = fxy = f12
∂y ∂x ∂y ∂x
∂2f ∂ ∂f
= = fyx = f21
∂x ∂y ∂x ∂y
∂2f ∂ ∂f
= = fyy = f22
∂y 2 ∂y ∂y

29. Don’t worry about the mixed partials
The “mixed partials” bookkeeping may seem scary. However, we
are saved by:
Theorem (Clairaut’s Theorem/Young’s Theorem)
If f is defined near (a, b) and f12 and f21 are continuous at (a, b),
then
f12 (a, b) = f21 (a, b).

30. Don’t worry about the mixed partials
The “mixed partials” bookkeeping may seem scary. However, we
are saved by:
Theorem (Clairaut’s Theorem/Young’s Theorem)
If f is defined near (a, b) and f12 and f21 are continuous at (a, b),
then
f12 (a, b) = f21 (a, b).
The upshot is that we needn’t worry about the ordering.

31. Example (Continued)
Let f (x, y ) = x 3 − 3xy 2 . Find the second derivatives of f .

32. Example (Continued)
Let f (x, y ) = x 3 − 3xy 2 . Find the second derivatives of f .
Solution
We have
f11 = (3x 2 − 3y 2 )x = 6x
f12 = (3x 2 − 3y 2 )y = −6y
f21 = (−6xy )x = −6y
f22 = (−6xy )y = −6x

33. Example (Continued)
Let f (x, y ) = x 3 − 3xy 2 . Find the second derivatives of f .
Solution
We have
f11 = (3x 2 − 3y 2 )x = 6x
f12 = (3x 2 − 3y 2 )y = −6y
f21 = (−6xy )x = −6y
f22 = (−6xy )y = −6x
Notice that f21 = f12 , as predicted by Clairaut (everything is a
polynomial here so there are no concerns about continuity). The
fact that f11 = −f22 is a coincidence.

34. Worksheet Problems 5–6

36. Outline
Partial Derivatives
Motivation
Definition
Other Notations
Worksheet
Second derivatives
Don’t worry about the mixed partials
Marginal Quantities

37. Marginal Quantities
If a variable u depends on some quantity x, the amount that u
changes by a unit increment in x is called the marginal u of x.
For instance, the demand q for a quantity is usually assumed to
depend on several things, including price p, and also perhaps
income I . If we use a nonlinear function such as
q(p, I ) = p −2 + I
to model demand, then the marginal demand of price is
∂q
= −2p −3
∂p
Similarly, the marginal demand of income is
∂q
=1
∂I

38. A point to ponder
The act of fixing all variables and varying only one is the
mathematical formulation of the ceteris paribus (“all other things
being equal”) motto.