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

 inverts the order of the numerator, subtracting  from .

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

 fails to square the denominator.

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  gives

 inverts the order of the numerator, subtracting  from .

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

 fails to square the denominator.

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

 inverts the order of the numerator, subtracting  from .

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

 fails to square the denominator.

The Chain Rule applies when differentiating compositions of functions.  Here,  equals the composition of two functions,  and . Let  and .  Then  and the Chain Rule tells us to differentiate the outside function  and multiply the result by the derivative of the inside function .  In other words, . Note that the inside function  is left untouched when the outside function  is differentiated.  Here,  and .  Remember, roots can (and should) be rewritten as fractional exponents, so  becomes  which is then differentiated like any other exponent.  So

 is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

 is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

 is a misapplication of the Power Rule which fails to subtract 1 from the original exponent.

The Chain Rule applies when differentiating compositions of functions.  Here,  equals the composition of two functions,  and . Let  and .  Then  and the Chain Rule tells us to differentiate the outside function  and multiply the result by the derivative of the inside function .  In other words, . Note that the inside function  is left untouched when the outside function  is differentiated.  Here,  and , so  which simplifies to .

 is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

 is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

 is a misapplication of the Chain Rule which substitutes the derivative of the inside function for the original inside function rather than multiplying the derivative of the outside function by the derivative of the inside function.

The Chain Rule applies when differentiating compositions of functions.  Here,  equals the composition of two functions,  and . Let  and .  Then  and the Chain Rule tells us to differentiate the outside function  and multiply the result by the derivative of the inside function .  In other words, . Note that the inside function  is left untouched when the outside function  is differentiated.  Here,  and , so  which simplifies to .

 is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

 is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

 is a misapplication of the Chain Rule which fails to preserve the original inside function when differentiating the outside function.

The Chain Rule applies when differentiating compositions of functions.  Here,  equals the composition of two functions,  and . Let  and .  Then  and the Chain Rule tells us to differentiate the outside function  and multiply the result by the derivative of the inside function .  In other words, . Note that the inside function  is left untouched when the outside function  is differentiated.  Here,  and , so  which simplifies to .

 is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

 is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

 is a misapplication of the Chain Rule which adds the derivative of the outside function to an incorrect derivation of the inside function.

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: