Math 18 Business Calculus
Chapter 8: Functions of Several Variables
Section 8.4: Computing Partial Derivatives Algebraically
Jennifer Cass, Instructor
As we did with our tables and graphs, we keep all but one variable fixed. In our typical functions with two input
variables, this means we fix one variable and treat it as though it were a constant. Then we apply our usual
derivative rules to the other variable.
Recall:
[ x n ]′ = nx n −1
[u n ]′ = nu n −1 ⋅ u ′
[e x ]′ = e x
[eu ]′ = eu ⋅ u ′
[ln x]′ =
1
x
[ln u ]′ =
1
⋅ u′
u
Examples (p.380)
# 2: Find
# 12: Find
#14: If
f x and f y if f ( x, y ) = 2 x 2 + 3 y 2 .
∂P
rt
if P = 100e .
∂r
f ( x, y ) = x3 + 3 y 2 , find f (1, 2) , f x (1, 2) , and f y (1, 2) .
Second-order partial derivatives arise when we take partial derivatives of our partial derivatives. When we have
two input variables we get a total of four second-order partial derivatives. It’s important to keep track of which
variables change at each level – the notation can help with this:
Notation:
∂ 2 z ∂  ∂z 
=   OR
∂x 2 ∂x  ∂x 
f xx = ( f x ) x
∂ 2 z ∂  ∂z 
=   OR
∂y 2 ∂y  ∂y 
∂2 z
∂  ∂z 
=   OR
∂y∂x ∂y  ∂x 
f xy = ( f x ) y
∂2 z
∂  ∂z 
=   OR
∂x∂y ∂x  ∂y 
Handy tip: The two “mixed partials”
f yy = ( f y ) y
f yx = ( f y ) x
f xy and f yx are always equal!
Examples (p.381) Calculate all four second-order partial derivatives and confirm the mixed partials are equal.
# 22:
f ( x, y ) = xe y [Start with the regular partial derivatives.]
# 28:
f ( x, t ) = t 3 − 4 x 2 t