Consider the following different definition of the a derivative:
where means .
- Find formulas for the “derivative” of the sum, difference, product, and quotient of functions.
- Find an expression for in terms of the usual derivative .
- Find the functions for which .
- Sum. Using the formula in the alternative definition,
Now, for the remaining limit term we multiply by (thanks to abcd in the comments for the suggestion!). So, we then have
Now, for the remaining limit, we use the product rule for the derivative in this alternative definition, which we derive below. (This last limit is . Below we derive that .) Therefore, we have
Difference. This follows just as the above derivation for except that we get minus signs:
Product. Using the alternative definition of the “derivative” we compute,
Quotient. Again, we start with the alternative definition and compute,
- Expressing in terms of the usual derivative we have,
We used the same idea we used in deriving some of the above formulas that , and that is a constant in , so we can pull it out of the limit.
- Now, to find the function such that we set the two expressions equal and solve for (using from part (c)),
In either case we have for some constant .