TITLE: Partial Derivatives
# We want to apply the concept of derivative to a function
  of several variables. Suppose f is a function of x and y.
  If we let only x vary while keeping y=b fixed at the value b
  we have a function of a single variable: <br>
	<center> g(x) = f(x,b) </center>
  If the derivative of g exists at x=a we call it the 
  <em> partial derivative of f with respect to x at (a,b)</em>
  and we write it by either fx(a,b) or D1f(a,b). We have
> #

                    df                  f(a + t, b) - f(a, b)
                   ---(a,b)  = Limit    ---------------------
                    dx         t -> 0           t

# The partial derivative of f w.r.t. x can be seen as the rate of change of the 
  value of the function with respect to a change in x when y is kept constant.
  If we visualize the graph of the function z = f(x,y) as a surface above
  the xy plane then the partial derivative of f w.r. to x at (a,b) is the
  slope of the line of intersection of the surface with the plane y=b at
  the point (a,b,f(a,b)). The following picture shows a surface z=f(x,y)
  and the graphs of the two curves z=g(x)=f(x,b) and z=h(y)=f(a,y),
!partial.gif
# The slope of the tangent line to z=g(x) at x=a is the partial
  derivative of f w.r.t. x at (a,b) and the slope of the tangent
  line to z=h(y) at y=b is the partial derivative of f w.r.t. y
  at (a,b). To compute partial derivatives all you have to do is
  to compute regular derivatives assuming all (but the variable
  you are computing the derivative w.r.t.) the other variables
  as if they were constants. Here are several examples with maple:
> f1 := (x,y) -> 4*x^2*y-sin(3*y);

                                          2
                        f1 := (x,y) -> 4 x  y - sin(3 y)

> Diff('f1(x,y)',x) = diff(f1(x,y),x);

                               d
                             ---- f1(x, y) = 8 x y
                              dx

> Diff('f1(x,y)',y) = diff(f1(x,y),y);

                         d                2
                       ---- f1(x, y) = 4 x  - 3 cos(3 y)
                        dy

# Higher order partials are defined by just taking the derivatives
  of derivatives.....<a href="examples-m2h.html">Look at this examples</a>
>