AP Calculus AB Exam Question:

Find the derivative of the equation: x^2 + xy + y^2 = 4 using implicit differentiation.

Solution:

To find the derivative of the given equation, we will apply the concept of implicit differentiation. Implicit differentiation allows us to calculate the derivative of an equation where both x and y are variables.

Step 1: Differentiate both sides of the equation with respect to x.

Differentiating with respect to x, we get:

2x + (x * dy/dx) + 2y(dy/dx) = 0

Step 2: Now we need to solve for dy/dx, which represents the derivative of y with respect to x.

Let's isolate the terms with dy/dx on one side:

(x * dy/dx) + 2y(dy/dx) = -2x

Step 3: Factor out dy/dx:

dy/dx(x + 2y) = -2x

Step 4: Finally, divide both sides by (x + 2y) to solve for dy/dx:

dy/dx = -2x / (x + 2y)

Therefore, the derivative of the equation x^2 + xy + y^2 = 4 with respect to x can be written as:

dy/dx = -2x / (x + 2y)

This is the desired solution obtained using implicit differentiation.