8.2 Partial Derivatives
Objective:
1. Find the first partial derivatives a function.
2. Evaluate the first partial derivatives a function at a given point.
3. Find the second-order partial derivatives a function.
For a function f ( x) of one variable x, there is no
ambiguity when we speak about the rate of change of
f ( x) with respect to x. However, the situation
becomes more complicated when we study the rate of
change of a function of two or more variables; f ( x, y ) .
We have infinitely many tangent lines at one point.
We will restrict ourselves to studying the rate of
change of the function f ( x, y ) at a point (a, b) in each
of two preferred directions—namely, the direction
parallel to the x-axis and y-axis.
This diagram shows the derivative of f ( x, y ) with
respect to x at x = a . We keep y fixed at a constant and
differentiate with respect to x—first partial derivative
of f with respect to x.
Notation:
δf
or
fx
δx
This diagram shows the derivative of f ( x, y ) with
respect to y at y = b . We keep x fixed at a constant and
differentiate with respect to y—first partial derivative
of f with respect to y.
Notation:
δf
or
fy
δy
Find the first partial derivatives of the function:
f ( x, y ) = 2 x 2 + 4 y + 1
f ( x, y ) =
x− y
x+ y
f ( x, y ) = ( x 2 + y 2 )
f ( x, y ) = e x lny
2
3
f ( x, y, z ) = xyz + xy 2 + yz 2 + zx 2
Evaluate the first partial derivatives of the function at the given point:
f ( x, y ) = x y + y 2 ; ( 2,1)
f ( x, y ) = e xy ; (1,1)
Second-Order Partial Derivatives
We may differentiate each of the first partial derivatives to obtain the second-order
partial derivatives of the function.
Differentiating the function f x with respect to x leads to the second partial derivative
∂2 f
∂
f xx = 2 = ( f x )
∂x
∂x
However, differentiating the function f x with respect to y leads to the second partial
derivative
∂2 f
∂
f xy =
= ( fx )
∂x∂y ∂y
Similarly, differentiation of the function f y with respect to x and with respect to y leads
to the second partial derivative
∂2 f
∂
∂2 f
∂
f yx =
= ( f y ) and f yy = 2 = ( f y )
∂x∂y ∂x
∂y
∂y
Find the second-order partial derivatives a function. In each case, show that the mixed
partial derivatives f xy and f yx are equal.
f ( x, y ) = x 2 y + xy 3
f ( x, y ) = ln(2 x + y 2 )