Product and Quotient Rule

Product and Quotient Rule - Product and Quotient Rule In...

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Product and Quotient Rule In the previous section we noted that we had to be careful when differentiating products or quotients. It’s now time to look at products and quotients and see why. First let’s take a look at why we have to be careful with products and quotients. Suppose that we have the two functions and . Let’s start by computing the derivative of the product of these two functions. This is easy enough to do directly. Remember that on occasion we will drop the (x) part on the functions to simplify notation somewhat. We’ve done that in the work above. Now, let’s try the following. So, we can very quickly see that. In other words, the derivative of a product is not the product of the derivatives. Using the same functions we can do the same thing for quotients.
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So, again we can see that, To differentiate products and quotients we have the Product Rule and the Quotient Rule . Product Rule If the two functions f(x) and g(x) are differentiable ( i.e. the derivative exist) then the product is differentiable and, The proof of the Product Rule is shown in the Proof of Various Derivative Formulas section of the Extras chapter. Quotient Rule If the two functions f(x) and g(x) are differentiable ( i.e. the derivative exist) then the quotient is differentiable and, Note that the numerator of the quotient rule is very similar to the product rule so be careful to not mix the two up! The proof of the Quotient Rule is shown in the Proof of Various Derivative Formulas section of the Extras chapter.
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