Question 1189629
.
Find the value of (f ∘ g)′ at the given value of x.
f(u)= u/ ((u^2)-1) ,u=g(x)=(6x^2)+x+4, x=0
~~~~~~~~~~~~~~~~~



        First, calculations in the post by @MathLover are  INCORRECT,  since she incorrectly differentiates the fractional function.


        So,  I came to make everything correctly.


        But it is not the only reason why I came.


        The entire solution can be done  MUCH  EASIER,  if you not only know the formal definition, 
        but,  in addition,  do understand its meaning and are trained to work with it . . . 


        You do not need to derive many-store long formulas for the composition of functions.


        See below,  how to make it in a simple way.



<pre>
The basic formula for the derivative of a composition function is the chain rule

              (f o g)'(x) = f'(g(x)) * g'(x).


    (1)  So, we first calculate g'(x) at the given point x= 0.

    (2)  Then we calculate g(x) at this point.

    (3)  Then we derive the formula for f'(u) 

    (4)  Finally, we calculate f'(u) at u = g(x) (substituting the value u= g(x), which we just have calculated in (2) )


Below is my implementation of this procedure.


(1)  g'(x) = 2*6*x + 1 = substitute x= 0 to get = 2*6*0 + 1 = 1.


(2)  g(0) = (6*0^2) + 0 + 4 = 0 + 0 + 4 = 4.


(3)  f'(u) = use the rule of differentiating the fractional function  f(u) = {{{u/(u^2-1)}}}    

         The numerator of the derivative will be  1*(u^2-1) - u*2u = -u^2-1. 

             We should calculate it at u= g(0) = 4, so the numerator's numerical value is -4^2-1 = -16-1 = -17.


         The denominator of the derivative will be (u^2-1)^2.

            We should calculate it at u= g(0) = 4, so the denominator's numerical value is (4^2-1)^2 = (16-1)^2 = 15^2 = 225.


(4)  Finally, (f o g)'(x) = f'(g(x)) * g'(x) at x= 0  is  {{{(-17/225)*1}}} = {{{-17/225}}}.    <U>ANSWER</U>.
</pre>

Solved.  &nbsp;&nbsp;&nbsp;&nbsp;//  &nbsp;&nbsp;&nbsp;&nbsp;All calculations are completed.


---------------


<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The lesson to learn from my post is &nbsp;THIS :</H3>

When you need calculate the value of the derivative of the composite function at given point, &nbsp;do it step by step using the chain rule.


When the component functions are fractional functions, &nbsp;deriving/making the global expression 
for the composition first - - - is not the best way, and it DOES NOT FOLLOW to the chain rule.


You can easily make many errors on the way, &nbsp;&nbsp;&nbsp;&nbsp;and you &nbsp;&nbsp;&nbsp;&nbsp;CERTAINLY &nbsp;&nbsp;&nbsp;&nbsp;will do them.


Following the change rule, do as the change rule dictates you:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- (1) &nbsp;&nbsp;first calculate the number &nbsp;g(x0) &nbsp;at given value of &nbsp;x0;


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- (2) &nbsp;&nbsp;next find the derivative &nbsp;g'(x0) &nbsp;and evaluate it at the given value of &nbsp;x0;


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- (3) &nbsp;&nbsp;next find the derivative &nbsp;f'(u) &nbsp;AS &nbsp;A &nbsp;FUNCTUION &nbsp;of &nbsp;u;


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- (4) &nbsp;&nbsp;then evaluate &nbsp;f'(u) &nbsp;at &nbsp;&nbsp;u = g(x0);


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- (5) &nbsp;&nbsp;finally, multiplate two numbers  &nbsp;f'(g(x0)  &nbsp;and  &nbsp;g(x0).


That's all.



////////////



Ignore the post by @MathLover1, &nbsp;since it is &nbsp;WRONG &nbsp;BOTH &nbsp;technically and conceptually.