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The Chain Rule 

One of the important rules of differential calculus is the product rule, which shows how to differentiate a product of functions.  Let us consider the following functional product: 

The product rule then tells us that 

or in a slightly shorter notation 

The question is, how does the product rule generalise to arbitrary products of functions?  Let us begin by defining such a product of n functions as follows: 

On the basis of the result for a two-product function, we can hazard a guess: 

                         (A) 

That is, for an n-product function, there are n terms that differ according to which of the individual functions is differentiated with respect to x. 

To prove that this assertion is correct, we use proof by induction.  The two steps we need to follow to complete the proof by induction are as follows: 

1.  Demonstrate that the statement to be proved holds for the case n = 0 (or n = 1) 

2.  Demonstrate that if the statement to be proved holds for the case n, then it also holds for the case n + 1. 

In undertaking the first step, we will actually start with the n = 2 case and insist that our formula (A) be valid for values of n greater than or equal to 2.  For the case n = 2, it can clearly be seen that (A) reduces to the form shown at the start of this article. 

To undertake the second step, we define a function that corresponds to the n + 1 case, based on the function defined above for an n-product function: 

Now let’s try differentiating this, using the two-function product rule that we know to be correct.  Using our short notation, the derivative is: 

If we substitute in our expressions above for y and its derivatives on the right-hand side, we get the following: 

Now compare this expression with (A).  It is of exactly the same form as (A), except with an extra term for the extra function included in the product.  This is the result we seek.  It shows that if our expression (A) for the product rule holds for an n-function product, then it also holds for an n + 1 – product function. 

The proof by induction is complete.  We have satisfied both requirements of the proof by induction, and hence we have proved that 

is the correct form for the product rule of an n-product function, as required.

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