Example: Product Rule For The Derivative
In this example, the binary operation in question is multiplication (of functions). The usual product rule for the derivative taught in calculus states:
or in logarithmic derivative form
This can be generalized to a product of n functions. One has
or in logarithmic derivative form
In each of the n terms of the usual form, just one of the factors is a derivative; the others are not.
When this general fact is proved by mathematical induction, the n = 0 case is trivial, (since the empty product is 1, and the empty sum is 0). The n = 1 case is also trivial, And for each n ≥ 3, the case is easy to prove from the preceding n − 1 case. The real difficulty lies in the n = 2 case, which is why that is the one stated in the standard product rule.
Alternative way to look at this is to generalize (a monoid homomorphism) to .
Read more about this topic: Mathematical Induction, Variants, Building On n = 2
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