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

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

We now prove a similar theorem for products of real-valued functions

**Theorem: ***Let with be differentiable at . Then, given by is differentiable at and *

**Proof: **We play a similar game to the sum proof. Namely, consider given by and given by and note that . Now, is differentiable at since each coordinate function is differentiable at and is differentiable at since it’s mutlilinear. Thus, by the chain rule we have that is differentiable at and . But, using the same logic as for the previous theorem we have that

And from our corollary concerning the total derivative of multilinear functions we know that

where evidently . Thus, with this in mind we see that for every we have

We now compute the derivative of the inner product of two differentiable functions. In particular:

**Theorem: ***Let where be differentiable at . Then, the function given by (where is the usual inner product (i.e. dot product) on ) is differentiable at and *

**Proof: **We play the same old game. Namely define given by where are the coordinate functions of respectively. We know since each coordinate function is differentiable at so is and moreover

We also know from a past corollary that is differentiable everywhere and

(where of course we use shorthand notation meaning ). Noticing then that we may conclude from the chain rule that is differentiable at and . Thus,

since was arbitrary the conclusion follows.

**References:**

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

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