Find the following derivatives. (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{\partial x}{\partial s}=8 t \) (Type an expression using \( s \) and \( t \) as the variables.) \( z=e^{3 x+5 y}, x=8 s t \), and \( y=6 s+5 t \) \( \frac{\partial z}{\partial y}=5 e^{3 x+5 y} \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{\partial y}{\partial s}=6 \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial x}{\partial t}=8 s \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial y}{\partial t}=\square \) (Type an expression using \( s \) and \( t \) as the variables.) (Ty

Let's solve the derivatives step by step based on the given expressions. 1. **Given:** \[ \frac{\partial x}{\partial s} = 8t \] This expression is already in the required form using \( s \) and \( t \). 2. **Given:** \[ z = e^{3x + 5y}, \quad x = 8st, \quad y = 6s + 5t \] We need to find \( \frac{\partial z}{\partial y} \): \[ \frac{\partial z}{\partial y} = 5 e^{3x + 5y} \] This expression is also already in the required form using \( x \) and \( y \). 3. **Given:** \[ \frac{\partial y}{\partial s} = 6 \] This expression is already in the required form using \( s \) and \( t \). 4. **Given:** \[ \frac{\partial x}{\partial t} = 8s \] This expression is already in the required form using \( s \) and \( t \). 5. **Now, we need to find \( \frac{\partial y}{\partial t} \):** Given \( y = 6s + 5t \), we can differentiate \( y \) with respect to \( t \): \[ \frac{\partial y}{\partial t} = 5 \] This expression is in the required form using \( s \) and \( t \). Now, let's summarize the results: - \( \frac{\partial x}{\partial s} = 8t \) - \( \frac{\partial z}{\partial y} = 5 e^{3x + 5y} \) - \( \frac{\partial y}{\partial s} = 6 \) - \( \frac{\partial x}{\partial t} = 8s \) - \( \frac{\partial y}{\partial t} = 5 \) Thus, the final answers are: - \( \frac{\partial y}{\partial t} = 5 \) (in terms of \( s \) and \( t \)).