Simple Application of Product Rule

These are some simple examples of applications of the product rule and the quotient rule. In the first problem there is another way of finding the derivative: distributing the product. But using the product rule is faster.

First, using the product rule, we put:



So,



Expanding this product we get (this last step is not required, I just do it to check the answer later):



Now, using the direct approach, that is, exanding the product in the original function:



And taking the derivative of that:



And the two answers are the same.

Next there is a similar application of the quotient rule. However, as I said in the quotient rule page, I prefer to just use the product rule instead of memorizing another formula. We have:



We can write this as:



And now we apply the product rule plus the chain rule:



So we have:



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