Total Derivatives

(1) Definition (Total Derivatives) For a function of two variables , if and are given increments and , then the corresponding increment of is

The differentials and are independent variables; that is, they can be given any values. Then the differential , also called the total differential, is defined by

(2) Example (Total Derivatives) If , find the differential Further, if changes from to and changes from to , compare the values of and Which is easier to compute or ?

Solution. By definition,

Putting , , , and , we get

The increment of is

Notice that but is easier to compute.

(3) Definition (Total Derivatives) If and its partial derivatives and are defined on an open region containing the point and and are continuous at then

so that

.

(4) Example (Total Derivatives) Using linear approximation and differentials to find an approximate value for

    Solution. Consider the function and observe that we can easily calculate Therefore, we take and

Since

we have













This approximation is accurate to two decimal places.

(5) Example (Total Derivatives) Using linear approximation and differentials to find an approximate value for Use differentials to find an approximate value for

    Solution. Consider the function

and observe that we can easily calculate Therefore, we take and in



Since

and


we have













This approximation is accurate to two decimal places.

(6) Definition (Total Derivatives) If , then is differentiable at provided can be expressed in the form

where and as Additionally, is said to be differentiable in the region of the plane if is differentiable at each point in .

(7) Proposition (Total Derivatives) Let be a function of two variables with in the domain of

(i) If is differentiable at it is also continuous at .

(ii) If is a function of and , and , , are continuous in a disk centered at , then is differentiable at

(8) Example (Total Derivatives) Let

That is, the function has the value when is in the first quadrant and is elsewhere. Show that the partial derivatives and exist at the origin, but is not differentiable there.

    Solution. Since , we have

and similarly Thus, the partial derivatives both exist at the origin. If were differentiable at the origin, it would have to be continuous there. Thus, we can show that is not differentiable by showing that it is not continuous at   Toward this end, note that is 1 along the line in the first quadrant but it is if the approach is along the -axis. This means that the limit does not exist. Therefore, is not differentiable there. So the point is, is a nondifferentiable function for which and exist, or in other words, the word differentiable means more than just the partial derivatives exist because the existence of partial derivatives does not guarantees that a function is differentiable.

(9) Example (Total Derivatives) Show that is differentiable for all .

    Solution. Compute the partial derivatives

and

Because and are all polynomials in and they are continuous throughout the plane. Therefore, the sufficient condition for differentiability theorem assures us that must be differentiable for all and

(10) Example (Total Derivatives) Determine whether the function is differentiable at either or Explain why.

    Solution. The limit


does not exist because along we have

and along we have

Therefore, is not continuous at and thus is not differentiable at To show that is differentiable at we compute the partial derivatives,

Since these partial derivatives and are continuous on any open disk not containing   we conclude that is differentiable at any point except ; and in particular at

(11) Example (Total Derivatives) Determine whether the function


is differentiable at either or Explain why.

    Solution. The limit

does not exist because along we have

and along we have

Therefore, is not continuous at and thus is not differentiable at To show that is differentiable at we compute the partial derivatives,

and

Since these partial derivatives and are continuous on any open disk not containing   we conclude that is differentiable at any point except ; and in particular at

Cite this as:
Total Derivatives
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/total-derivatives.html