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The sum, product and quotient rules show us how to differentiate a great many different functions from the reals to the reals. However some functions, such as are a *composition* of functions, and these rules don’t tell us what the derivative of is. There is, however, a theorem called the *chain rule *that tells us how to differentiate these functions. Here we present the proof. In class we won’t prove this assertion but we will make one attempt to explain why it takes the form it does. In general only practise can make you proficient in the use of the chain rule. See http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf or any other textbook (such as a LC text book) with exercises.

**Proposition 4.1.8 (Chain Rule)**

*Let be functions, and let denote the composition (that is for each ). If such that is differentiable at and is differentiable at , then is differentiable at with*

Let . Their *composition* is the function is defined by

*Let and be functions with continuous at some point , and continuous at the point . Then is continuous at .*

**Proof**: *For each , we must find a such that*

Let , since is continuous at , :

But also is continuous at , so (*we can get -close to *), such that

So therefore,

Here we present the proof of assertions 3. and 4. of the following proposition. The proofs of 1. and 2. will be presented in class and here they are assumed. The proofs presented here will not be presented in class.

**Proposition 3.1.4 (Calculus of Limits)**

*Suppose that and are two functions , and that for some we have*

* , andÂ .*

*for some . Then*

*.**If , .**.**If , .**If , and then .**Read the rest of this entry »*

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