Question

Use the given equation 9y cos(x) = x² y² to find dy/dx.

Answer 1

To find dy/dx of 9y cos(x) = x² y², we use implicit differentiation and apply the product rule. Simplify the equation and isolate dy/dx, giving us (2x y² + 9y sin(x)) / (9 cos(x) - x² (2y)) as the derivative.

Step-by-step explanation:

To find the derivative dy/dx of the equation 9y cos(x) = x² y², we need to use implicit differentiation because y is a function of x and it is not isolated on one side of the equation. Here's a step-by-step process:

1. Differentiate both sides of the equation with respect to x.
2. Apply the product rule to y cos(x) which becomes y' cos(x) - y sin(x), where y' represents dy/dx.
3. Simplify the equation and solve for dy/dx.

Let's differentiate both sides:

9y' cos(x) - 9y sin(x) = 2x y² + x² (2y)y'

Rearrange terms to isolate y' on one side:

y' (9 cos(x) - x² (2y)) = 2x y² + 9y sin(x)

Solve for y':

y' = (2x y² + 9y sin(x)) / (9 cos(x) - x² (2y))

The derivative dy/dx for the given equation is therefore (2x y² + 9y sin(x)) / (9 cos(x) - x² (2y)).