How do you find dy/dx for the equation 3y cos(x) – x^2y^2?

To find dy/dx for the equation 3y cos(x) – x²y² = 0, we will use implicit differentiation.

First, we start with the equation:

3y cos(x) - x2y2 = 0

We will differentiate both sides with respect to x. Remember that when we differentiate terms involving y, we will need to apply the chain rule since y is a function of x.

Let’s differentiate each term:

• For the term 3y cos(x):
• Using the product rule:
  • Differentiate 3y: 3(dy/dx).
  • Differentiate cos(x): -sin(x).
  • The derivative is: 3(dy/dx) cos(x) + 3y(-sin(x)).
• For the term -x²y²:
• Using the product rule again:
  • The derivative is: -2xy² – x²(2y)(dy/dx).

Setting the derivative equal to zero gives us:

3(dy/dx) cos(x) - 3y sin(x) - 2xy2 - 2x2y(dy/dx) = 0

We can now group the terms involving dy/dx

(3 cos(x) - 2x2y)(dy/dx) = 3y sin(x) + 2xy2

dy/dx = (3y sin(x) + 2xy2) / (3 cos(x) - 2x2y)