Partial Derivatives

Partial Derivatives Examples And A Quick Review of ImplicitDifferentiation
Given a multi-variable function, we defined the partial deri vative of one variable with respect to another
variable in class. All other variables are treated as consta nts.
Here are some basic examples:
1. If z= f(x, y ) = x4
y 3
+ 8 x2
y + y4
+ 5 x, then the partial derivatives are
∂ z
∂ x = 4
x3
y 3
+ 16 xy+ 5 (Note: yfixed, xindependent variable, zdependent variable)
∂ z
∂ y = 3
x4
y 2
+ 8 x2
+ 4 y3
(Note:xfixed, yindependent variable, zdependent variable)
2. If z= f(x, y ) = ( x2
+ y3
)10
+ ln( x), then the partial derivatives are
∂ z
∂ x = 20
x(x 2
+ y3
)9
+ 1 x (Note: We used the chain rule on the first term)
∂ z
∂ y = 30
y2
(x 2
+ y3
)9
(Note: Chain rule again, and second term has no y)
3. If z= f(x, y ) = xexy
, then the partial derivatives are
∂ z
∂ x =
exy
+xye xy
(Note: Product rule (and chain rule in the second term)
∂ z
∂ y =
x2
e xy
(Note: No product rule, but we did need the chain rule)
4. If w= f(x, y, z ) = y
x
+ y+ z, then the partial derivatives are
∂ w
∂ x =
(
x + y+ z)(0) −(1)( y) (x + y+ z