Learn about the product rule - when to use it and why
Given a function , how would you differentiate the function?
Differentiating it with the definition would be quite a chore. Just try expanding ...
Well, is the product of two functions, and , right?
Now think about what the derivative represents. The derivative describes a rate of change. How does the product change?
If we increase by some small quantity , it will cause a change in as well as in . The changes in the composite functions in turn depend on and . So we'd expect the terms and to crop up somewhere.
Since we're dealing with a product, it'd also be logical if there was some multiplication going on here.
Thus, the rule for the derivative of a product is:
It appears all the time, so you should be able to give the right answer.
then we get:
The product rule
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