TITLE: Computing partial derivatives
# !b2 Problem:
Compute the partial derivatives with respect to
x and y for 
> f := (x,y) -> sin(sqrt(x))*log(y^2);
                                                       2
                      f := (x, y) -> sin(sqrt(x)) log(y )

# and evaluate them at (1,1/2).

# !b3 Answer:
Just do it!
# w.r.t. x is fx given by
> fx := diff(f(x,y),x);
                                         1/2      2
                                    cos(x   ) ln(y )
                          fx := 1/2 ----------------
                                           1/2
                                          x

# and w.r.t. y is fy given by
> fy := diff(f(x,y),y);
                                            1/2
                                       sin(x   )
                               fy := 2 ---------
                                           y

# For the purposes of evaluation it is more convenient to use
the "D" operator. Hence, the value of the partial w.r.t. x evaluated
at (1,1/2) is given by,
> D[1](f)(1.,0.5);
                                 -.3745090200

# and fy(1,1/2) is given by,
> Dfy := unapply(fy,(x,y));
                                                 1/2
                                            sin(x   )
                         Dfy := (x, y) -> 2 ---------
                                                y

# Now apply,
> Dfy(1.,0.5);
                                  3.365883940


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