Product and quotient rule, Mathematics

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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  /x)′ =(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.


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