AP Calculus AB Exam Question:

Find dy/dx by using implicit differentiation for the equation: 6x^2 + 4xy + 3y^2 = 10.

Step-by-step Solution:

To find dy/dx using implicit differentiation, we need to differentiate each term of the equation with respect to x, treating y as a function of x. Let's go through each step in detail:

1. Start by differentiating each term of the equation with respect to x:

   • (d/dx)(6x^2) + (d/dx)(4xy) + (d/dx)(3y^2) = (d/dx)(10).
   • Simplify each term:
     • 12x + 4y + 6yy' = 0, since the derivative of a constant (10) is 0.

2. Now, isolate the derivative term dy/dx or y' by moving all the other terms to the other side of the equation:

   • 12x + 4y = -6yy'.

3. Rearrange the equation to solve for dy/dx:

   • Divide both sides by -6y:
     • (12x + 4y) / (-6y) = y'.
   • Simplify the expression using cancellation and rearrange:
     • -(2x + 2y/3) / y = y'.
     • Alternatively, we can write:
       • y' = -(2x + 2y/3) / y.

Thus, the derivative dy/dx or y' for the equation 6x^2 + 4xy + 3y^2 = 10 is -(2x + 2y/3) / y.