Ind \( \frac{d y}{d x} \) if \( x=a \sin ^{2} t \) and \( y=\operatorname{aros}^{2} t, a \) is \( n \) const

To find \( \frac{d y}{d x} \), we will need to use the chain rule. First, let's denote \( x = a \sin^2 t \) and \( y = \operatorname{aros}^2 t \). We start by differentiating both \( x \) and \( y \) with respect to \( t \): 1. For \( x \): \[ \frac{dx}{dt} = a \cdot 2 \sin t \cos t = a \sin(2t) \] 2. For \( y \): \[ \frac{dy}{dt} = 2 \operatorname{aros}(t) \cdot \frac{d (\operatorname{aros}(t))}{dt} \] Since \( \frac{d(\operatorname{aros}(t))}{dt} \) depends on the function you are considering, you may refer to the derivative of the inverse function you have. Now, to find \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2 \operatorname{aros}(t) \cdot \frac{d(\operatorname{aros}(t))}{dt}}{a \sin(2t)} \] So the complete expression depends on how you differentiate \( \operatorname{aros}(t) \). If we assume it's a hypothetical function with a known derivative, plug that into the formula above: \[ \frac{dy}{dx} = \frac{2 \operatorname{aros}(t) \cdot \text{[known derivative]}}{a \sin(2t)} \] Make sure to calculate and substitute the appropriate values based on the specifics of \( \operatorname{aros}(t) \) if they are defined in your problem context.