The Quotient Rule applies when differentiating quotients of functions.  Here,  equals the quotient of two functions,  and .  Let  and .  (Think:  is the "low" function or denominator and  is the "high" function or numerator.)  The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function.  In other words,

Here,  so .  Similarly,  so .

Then

Factoring out  from the numerator gives

Which simplifies to

 inverts the order of the numerator, subtracting  from .

 adds the products in the numerator, rather than subtracting them.

 fails to square the denominator.

This problem tests the knowledge of two concepts needed to compute the derivative of the function above – the chain rule and the quotient rule. The first step of the chain rule is the application of the power rule to the entire function, yielding the term:

To complete the chain rule, this term must then be multiplied by the derivative of only the function within the parentheses, which requires the application of the quotient rule. Remember the quotient rule is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function all over the bottom function squared. This yields the complete derivative of the function, with the first factor being the term above, and the second factor being the derivative of only the equation in parentheses:

Multiplying the equation by    and simplifying, this equation becomes that given by answer choice 

Applying implicit differentiation to the equation, we differentiate with respect to x, treating the variable y as a function of x:

Rearranging the equation and factoring out y', we get:

Finally, dividing and factoring out a 2y from the denominator gives us:

The critical points of the function are those at which its slope is 0, so to find the critical points we must first take the derivative of the function:

The points where the slope of the function is 0 are those where the derivative is 0, so we then factor the first term in the equation above to make it easier to see at what values of x the slope of the function is 0:

We can see from the equation for our derivative that only the first and second terms can make the slope equal zero, so we set each of these terms equal to zero and solve for x to find where our critical points would be:

Remember critical points are the coordinate pair  we would need to plug in our x values into the original function to find the cooresponding y values. However, this question only asks to find the x values of the critical points.

To take the derivative of this function lets first distribute the 4x through the binomial.

From here we use the power rule which states when,

.

We also need to remember,

.

Therefore our derivative becomes,

.

To find the derivative of the function, we must use the Chain Rule. Since  and  are both functions, we can 

Knowing this we can differentiate the function

We must use the Product Rule and the Chain Rule to find the derivative of this function. If we let

Now we can find the derivative of the two parts using the Chain Rule

Combining everything together gives us

To find the derivative of the function we must use the Chain Rule and the Quotient Rule.

The Quotient Rule is 

The Chain Rule is 

If  and 

Using the Chain Rule on the numerator of the function gives us,

Taking the derivative of the denominator gives us

Now we can combine all of the pieces into the Quotient Rule to find the derivative of the entire function

Simplifying this gives us

To find the derivative of this function, we must use the Product Rule and the Chain Rule.

The equation for the Product Rule is

The equation for the Chain Rule is 

Applying the Chain Rule to  gives

Using the Product Rule, we can now find the derivative of the entire function. If  and  then the derivative of the function will be,

To find the derivative of this equation, we must use the Chain Rule.

Applying this to the function we are given gives us