Derivatives Refresher for constrained dynamics assignment

1) function of one variable: f(x)=x+2
   - derivative of f with respect to x: f'(x)=1
   - second derivative of f with respect to x: f''(x)==0

2) given a vector V=(x,y,z)
   - The derivative of a vector must be the "time derivative"
     denoted V' = (x',y',z')
   - where each component derivative is also with respect to time

3) function of two variables: f(x,y)=2x+y^3
   - partial derivative w.r.t x is f'x=2
   - partial derivative w.r.t y is f'y=3y^2
   - second partial derivatives:
       f''x=0
       f''y=6y
   - gradient of f = (f'x,f'y) = (2,3y^2)
   - second partial vector of f = (f''x,f''y) = (0,6y)
   - time derivative of f = 2*x' + 3*y^2*y'
   - second time derivative of f = 2*x'' + 3*(2*y*y'*y' + y^2*y'')
   - gradient of the gradient is a Hessian matrix:

       [ D^2f/Dx^2  D^2f/DxDy ]
       [ D^2f/DyDx  D^2f/Dy^2 ]

     Because a gradient is a vector, and the gradient of a
     vector whose components are functions of more than one variable
     is a Hessian matrix:

     - gradient of f(x,y) is vector (Df/Dx,Df/Dy)
     - gradient of vector (Df/Dx,Df/Dy) which is still a function of two
       variables can be thought of as the same function: the gradient
       takes a function of more than one variable and creates vector
       whose components consist of the partial derivatives of each
       variable:

       gradient of vector (Df/Dx,Df/Dy) =

         [ Df/Dx(Df/Dx,Df/Dy) ]
         [ Df/Dy(Df/Dx,Df/Dy) ]

       which becomes the matrix:
 
         [ D^2f/Dx^2  D^2f/DxDy ]
         [ D^2f/DyDx  D^2f/Dy^2 ]

       I guess because the derivative of a vector is the component-
       wise derivatives??

4) vector-valued function of a vector X: f(X)=X*X
   where "*" is the dot-product
   - so time derivative of f: f'(X) = X'*X + X*X' = 2*X*X'
   - second time deriv of f: f''(X) = dfx(X'*X) + dfx(X*X')
                                    = X''*X + X'*X' + X'*X' + X*X''
                                    = 2*X*X'' + 2*X'*X'
                                    = 2*(X*X'' + X'*X')
   - this is still a function of only one variable so the
     partial derivative or gradient operator does not apply.