Question 3: How do you take the second partial derivative of a function?
In functions of one variable, the second derivative is simply the derivative of the first
derivative. For instance, if g ( x)  x3  2 x 2 then the first derivative is
g   x   3x 2  4 x
Taking another derivative yields
g   x   6 x  4 For functions of more than one variable, more variables mean more possibilities for the
first derivative. Suppose f  x, y   x3  x 2 y  y 3 . If y is held constant, the first derivative
with respect to x is
f x  x, y   3x 2  2 xy The second derivative may be taken with respect to x again to give
f xx  x, y   6 x  2 y Or we could take a partial derivative of f x  x, y  with respect to y to give
f xy  x, y   2 x The first partial derivative with respect to x yields two different second partial
derivatives. The first partial derivative with respect to y also yields two second partial
derivatives, f yy  x, y  and f yx  x, y  .
In total, there are four possible second partial derivatives for a function of two variables.
There are also several notations used to name them.
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Second Partial Derivatives
The function z  f  x, y  has four possible second partial
derivatives (if they exist). They are indicated by
f xx  x, y   z xx 
  z   2 z

x  x  x 2
f xy  x, y   z xy 
  z   2 z

y  x  yx
f yy  x, y   z yy 
  z   2 z

y  y  y 2
  z   2 z
f yx  x, y   z yx    
x  y  xy
For continuous functions, the partial derivatives f xy  x, y  and f yx  x, y  are equal.
Example 7
Second Partial Derivatives
For the function f  x, y   x3  x 2 y  y 3 , find f yy  x, y  and f yx  x, y  .
Solution If x is held constant, the first derivative with respect to y is
f y  x, y   x 2  3 y 2
Remember, since x is constant the partial derivative of x3 is zero. If we
continue to hold x constant and take another partial derivative with
respect to y, we get
f yy  x, y   6 y
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If we hold y constant in f y  x, y  and take the partial derivaitve with
respect to x, we get
f yx  x, y   2 x
Note that f yx  x, y   f xy  x, y  as expected.
Example 8
Second Partial Derivatives
If z  ln( xy )  x 2 y 4 , find all second partial derivatives.
Solution The partial derivative with respect to x is
z 1
  y  2 xy 2
x xy

1
 2 xy 2
x
where the Chain Rule is used to find the derivative of the first term. If
we take the derivative with respect to x again, we get
2 z
1
  2  2 y2
2
x
x
If we take the derivative with respect to y, we get
2 z
 4y
yx
The first partial derivative with respect to y is
z 1
  x  4 xy
y xy

1
 4 xy
y
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Taking another partial derivative with respect to y gives
2 z
1
  2  4x
2
y
y
Taking a partial derivative with respect to x of the first partial derivative
with respect to y leads to
2 z
 4y
xy
For functions of one variable, the second derivative is a tool which may be used to
determine whether a critical number corresponds to a relative maximum or relative
minimum. Second partial derivatives play a similar role in functions of two variables. In
the next section we will use them to determine whether a surface has a relative
maximum or minimum.
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