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Differentiability

A function of one variable is differentiable at a point if its derivative has a value at that point. However, the term differentiable is not used the same for functions of several variables; in particular, a function of several variables is not necessarily differentiable if its partial derivatives have values at a given point. With functions of two variables, a function having a tangent plane is equivalent to a function being differentiable, and similarly for functions of several-variables. As is the case for functions of one variable, several-variable functions that are differentiable are also continuous. This topic illustrates the sufficient conditions for differentiability; techniques for finding the equation of a tangent plane; and using differentials for approximations.

Definition (Differentiability) 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 .

Proposition (Differentiability) 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

    Proof. (i) We wish to show that as or equivalently, that

If we set and and let denote the increment of at we have,

Then because as we wish to prove that Since is differentiable at we have

where and as It follows that

as desired.

Example (Differentiability) 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.

Example (Differentiability) 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

Example (Differentiability) 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

Example (Differentiability) 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:
Differentiability
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/differentiability.html

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