How to find dy/dx using the equation 4y cos(x) + x^2y^2?

To find dy/dx from the equation 4y cos(x) + x²y² = 0, we will use implicit differentiation.

First, let’s differentiate both sides of the equation with respect to x:

d/dx (4y cos(x) + x²y²) = 0

Now, apply the product rule and chain rule as necessary:

• For the term 4y cos(x):
• Using the product rule:
• 4( dy/dx * cos(x) + y * (-sin(x)))
• For the term x²y²:
• Again, applying the product rule:
• (2xy² + x²(2y dy/dx))

Now, substituting these back, we get:

4(cos(x) dy/dx – y sin(x)) + 2xy² + 2x²y dy/dx = 0

Next, we’ll collect the terms involving dy/dx on one side:

4cos(x) dy/dx + 2x²y dy/dx = ysin(x) – 2xy²

Now, factor out dy/dx:

dy/dx(4cos(x) + 2x²y) = ysin(x) – 2xy²

Finally, solve for dy/dx:

dy/dx = (ysin(x) – 2xy²) / (4cos(x) + 2x²y)

This is the desired result for dy/dx using the given equation.