product and quotient rule firstly lets see why

Question

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.