## Further Properties of the Total Derivative (Pt. II)

**Point of Post: **This is a continuation of this post.

We showed all of these proofs because it’s helpful to go through them, but they are all generalized by the following theorem:

**Theorem: ***Let , where , be differentiable at and let be multilinear then given by is differentiable at and*

**Proof: **Unsurprisingly we mimic the special cases we’ve already done. In particular, we define by

(where is the coordinate function of ). We know that is differentiable at since each of its coordinate functions is differentiable at . Also, we know that is differentiable since it’s multilinear. Thus, since we know from the chain rule that is differentiable at and . But, as always

and we know that

where . It then follows that for any one has

since was arbitrary the conclusion follows.

As a last point in this post we generalize the ‘quotient rule’ to mappings . In other words:

**Theorem: ***Let where be differentiable at with , then is differentiable at and*

**Proof: **We begin by noting that since is continuous at we have that implies there exists an open ball of for which is non-zero. Clearly then choosing implies that . Consequently it makes sense to consider limits as with in them. Thus, consider then that for one has that

so that for

(where is the operator norm )but since both terms go to zero, the first since the numerator goes to zero while the denominator goes to something non-zero and the second because it’s a term going to a non-zero constant times a term which goes to zero by the definition of differentiability, we may conclude that is in fact differentiable at and has the claimed derivative.

Using the product rule we evidently get the following corollary:

**Corollary: ***Let where be differentiable at . If then is differentiable at and $*

**References:**

1. Spivak, Michael. *Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus.* New York: W.A. Benjamin, 1965. Print.

[…] first and last one we have proven before (here and here respectively) and the second can be proved thinking of multiplication as a bilinear map and […]

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