**MATH6037 please skip to the end of this entry.**

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*

*Proof: *Define a function by

Then , since we assumed that is differentiable at . Hence is continuous at . Rearranging the equation above we see that

(*)

Now for any ,

where we define for all . The right hand side of this last equation is of the form of the left hand side of (*), with , and so

But is differentiable at , hence continuous there as well, thus

and

That is, is a function on that is continuous at and satisfies , and is continuous at and satisfies . Hence is continuous at , with by Proposition 3.3.5. So now

as required

# The Chain Rule… by Rule

Suppose is some function with derivative . :The following is a nice, rough and ready definition of the chain rule:

*Let where is a function with derivative . Then has derivatative *.

Translating this into some common examples (note that you are expected to know the derivatives of the ‘outside’ functions):

**Logarithms:**

Suppose . Now has derivative , hence:

.

**Inverse Trigonometric**

Suppose , then:

.

Suppose that , then:

.

**Powers**

Suppose , then

.

**Trignometric**

Suppose , then

.

Suppose , then

.

Suppose , then

.

**Exponential**

Suppose that , then

.

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November 2, 2011 at 1:12 pm

MS2001: Week 7 « J.P. McCarthy: Math Page[…] The proof of the chain rule and the “chain rule by rule” here. […]