We know that: The derivative of the difference of two functions is the difference of the derivatives of the two functions. This was called the Difference Rule, and it is from the previous lesson. We might guess that the same type of thing happens with products. Is it true that the product of the derivatives is equal to the derivative of the product? We see from the following example, however, that this does not work.
Example.





functions  Here we have the two functions we have chosen, their product, and the derivative of the product.  
derivatives 



Here we have the two derivatives and their product. Notice that this is not equivalent to the cell above. 
Definition.
Product Rule  The derivative of a product is the first function multiplied by the
derivative of the second function plus the second function multiplied by
the derivative of the first fuction. In symbols, this is shown below:

Now armed with the proper rule, let's finish our example from above.

Here is our new product rule. 

On the left side we first multiply the two functions in preparation for differentiating in the next step. 

Differentiate the left side and simplify the right side. 

Finish simplifying the right side. Now we can see how the product rule is needed to differentiate a product. 
Definition.
Quotient Rule  The derivative of a quotient is the denominator times the derivative
of the numerator minus the numerator times the derivative of the denominator,
all divided by the square of the denominator. In symbols, this is shown
below:

Here again, there is no short cut intuitive method here. When you need tod ifferentiate a quotient of two functions, you must follow the quotient rule.
Example.

Given sample problem. 

Here we use our new quotient rule. 

Do the products and simplify. 

Simplify further. This is our derivative. 
