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). 

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