How can I find the derivative of y with respect to x using implicit differentiation for the equation x^2 + 4xy + y^2 = 4?

by

To find the derivative of y with respect to x using implicit differentiation for the equation:

x^2 + 4xy + y^2 = 4

we first need to differentiate both sides of the equation with respect to x. Remember that any term containing y will require the application of the chain rule since y is a function of x.

Let’s differentiate each term:

  • d/dx(x^2) = 2x
  • d/dx(4xy) = 4(y + x(dy/dx)) (using the product rule)
  • d/dx(y^2) = 2y(dy/dx)

Now, let’s differentiate the right side:

  • d/dx(4) = 0

Putting it all together, we get:

2x + 4(y + x(dy/dx)) + 2y(dy/dx) = 0

Now, simplify the equation:

2x + 4y + 4x(dy/dx) + 2y(dy/dx) = 0

Combine the terms with dy/dx:

4x(dy/dx) + 2y(dy/dx) = -2x - 4y

Factoring out dy/dx:

dy/dx(4x + 2y) = -2x - 4y

Finally, solve for dy/dx:

dy/dx = (-2x - 4y) / (4x + 2y)

This expression gives you the derivative of y with respect to x based on the implicit relationship defined by the original equation.

Leave a Comment