Product and Quotient Rule : Firstly let's see why we have to be careful with products & quotients. Assume that we have the two functions f ( x ) = x3 and g ( x ) = x6 . Let's begin by calculating the derivative of the product of these two functions. It is easy enough to do directly.
( f g )′ = ( x3 x6 )′ =( x9 )′ = 9x8
Recall that on occasion we will drop the (x) part on the functions to simplify notation somewhat. We've done this in the work above.
Now, let's attempt the following.
f ′ ( x ) g′ ( x ) = (3x2 )(6x5 ) = 18x7
Thus, we can very rapidly see that.
( f g )′ ≠ f ′ g ′
In other terms, the derivative of a product is not the product of the derivatives.
By using the same functions we can do the similar thing for quotients.
( f /g)′ = (x3 /x6 )′ =(1/x3)′ = (x-3)′ = -3x-4 = - 3/x4
f ′ (x )/g'(x) = 3x2 /6x5 = 1/2x3
Hence, again we can see that,
(f/g)'≠f'/g'
To differentiate products & quotients we have the Product Rule & the Quotient Rule.