Question
ind the following derivatives. \( z_{s} \) and \( z_{\text {}} \), where \( z=e^{3 x+5 y}, x=8 s t \), and \( y=6 s+5 t \) \( \frac{\partial z}{\partial x}=3 e^{3 x+5 y} \) (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.) \( \frac{\partial z}{\partial y}=5 e^{3 x+5 y} \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{\partial y}{\partial s}=\square \) (Type an expression using \( s \) and \( t \) as the variables.)
Ask by Moran Weaver. in the United States
Mar 17,2025
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Answer
\( \frac{\partial z}{\partial x} = 3e^{3x+5y} \), \( \frac{\partial x}{\partial s} = 8t \), \( \frac{\partial z}{\partial y} = 5e^{3x+5y} \), \( \frac{\partial y}{\partial s} = 6 \)
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The derivative \( \frac{\partial y}{\partial s} \) can be found from the expression for \( y \): Given that \( y = 6s + 5t \), the derivative with respect to \( s \) is: \[ \frac{\partial y}{\partial s} = 6 \] Now let's summarize the derivatives you've found: 1. \( \frac{\partial z}{\partial x} = 3 e^{3x + 5y} \) 2. \( \frac{\partial x}{\partial s} = 8t \) 3. \( \frac{\partial z}{\partial y} = 5 e^{3x + 5y} \) 4. \( \frac{\partial y}{\partial s} = 6 \)