Part 1: The Total Derivative

To this point, we have considered only partial derivatives of a function f( x,y) . In this section, we introduce the concept of a total derivative of a transformation.  

To begin with, let us denote f( x,y) by f( x) , where x = á x,y ñ and let p denote the point ( p,q).  If we define

Df = f( x) -f( p)     and   Dx = x-p

then the concept of a total derivative is captured by the idea that there is a vector m such that

Df » m·Dx

(1)

If we denote the total derivative m by Ñf( p) , where Ñ is a symbol called "del," then (1) is equivalent to

Df-Ñf( p) ·Dx » 0

(2)

The definition of the total derivative then follows once we define exactly what äpproximately zero" means in (2). 

Definition 5.1: A function f( x,y) is differentiable at a point ( p,q) if there exists a vector Ñf( p) for which

lim Dx® 0     | Df-Ñf( p) ·Dx| || Dx || = 0

(3)

When it exists, Ñf( p) is the total derivative of f( x) at p.

The vector Ñf( p) is also called the gradient of f( x) at p.

If we write Ñf( p) = á a,b ñ , then we can determine the values of a and b by evaluating the limit in (3) in two different directions.  If Dx = áh,0 ñ , then

Df = f( p+h,q) -f( p,q)

and Ñf( p) ·Dx = áa,b ñ · á h,0 ñ = ah. Also, || Dx || = |h|, so that (3) becomes

lim h® 0     | f( p+h,q) -f(p,q) -ah| | h |     = 0   ê ê   lim h® 0    æ è    f( p+h,q) -f(p,q) h ö ø  - a  ê ê       = 0 | fx( p,q) - a |  = 0

which implies that a = f_x( p,q) .  Similarly, it can be shown that b = f_y( p,q) , so that

Ñf( p,q) = á fx( p,q) ,fy(p,q) ñ

This yields the following:    

Theorem 5.2: If f( x,y) is differentiable at a point ( p,q) , then its total derivative is given by the gradient of f at ( p,q) , which is

Ñf( p,q) = á fx( p,q) ,fy(p,q) ñ

The total derivative is also known as the Jacobian of the mapping f( x,y) for reasons that will become apparent in the next chapter. 

EXAMPLE 1    Find the total derivative (i.e., gradient) of

f( x,y) = x2+3xy

Solution:  Since f_x = 2x+3y and f_y = 3x, the total derivative is

Ñf = á 2x+3y,3x ñ

Definition 5.1 can be applied to a function f of any number of variables, in which case Theorem 5.2 says that Ñf is the vector of first partial derivatives. For example, the gradient of a function of 3 variables U( x,y,z) is given by

ÑU = á Ux,Uy,Uz ñ

and ÑU is the total derivative of U( x,y,z) .

  Check your Reading: Why was the product rule not used in evaluating  f_x(x,y) in example 1?