Given a
function *f* of *n* variables ,
we define the *partial derivative* relative to variable , written
as , to be the derivative of
*f* with respect to treating all variables except as
constant.

The answer is: .

Let *x* denote the vector .
With this notation, ,
, etc.
The *gradient* of *f* at *x*, written , is the vector
. The gradient vector
gives the direction of steepest ascent of the function *f*
at point *x*.
The gradient acts like the derivative in that small changes around
a given point can be estimated using the gradient.

where denotes the vector of changes.

In this case, and . Since and , we get

.

So .

**Hessian matrix**

Second partials
are obtained from *f*(*x*) by taking
the derivative relative to (this yields the
first partial )
and then by taking the derivative of
relative to . So we can compute
,
and so on. These values are arranged into the *Hessian* matrix

The Hessian matrix is a symmetric matrix, that is .

**Example 1.1.1 (continued):**
*Find the Hessian matrix of
*

The answer is

Mon Aug 24 14:09:40 EDT 1998