ind the following derivatives. $$ z_{s} $$ and $$ z_{ ext {}} $$, 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.)

Question

ind the following derivatives. $$ z_{s} $$ and $$ z_{	ext {}} $$, 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.)

UpStudy AI · Accepted Answer

To find the derivatives $$ \frac{\partial z}{\partial x} $$, $$ \frac{\partial x}{\partial s} $$, $$ \frac{\partial z}{\partial y} $$, and $$ \frac{\partial y}{\partial s} $$, we will use the chain rule and the given expressions for $$ z $$, $$ x $$, and $$ y $$.

Given:
$$ z = e^{3x+5y} $$
$$ x = 8st $$
$$ y = 6s + 5t $$

1. Find $$ \frac{\partial z}{\partial x} $$:
Using the chain rule, we have:
$$ \frac{\partial z}{\partial x} = 3e^{3x+5y} $$

2. Find $$ \frac{\partial x}{\partial s} $$:
Differentiating $$ x = 8st $$ with respect to $$ s $$, we get:
$$ \frac{\partial x}{\partial s} = 8t $$

3. Find $$ \frac{\partial z}{\partial y} $$:
Using the chain rule, we have:
$$ \frac{\partial z}{\partial y} = 5e^{3x+5y} $$

4. Find $$ \frac{\partial y}{\partial s} $$:
Differentiating $$ y = 6s + 5t $$ with respect to $$ s $$, we get:
$$ \frac{\partial y}{\partial s} = 6 $$

Therefore, the derivatives are:
$$ \frac{\partial z}{\partial x} = 3e^{3x+5y} $$
$$ \frac{\partial x}{\partial s} = 8t $$
$$ \frac{\partial z}{\partial y} = 5e^{3x+5y} $$
$$ \frac{\partial y}{\partial s} = 6 $$
$$ \frac{\partial z}{\partial x} = 3e^{3x+5y} $$, $$ \frac{\partial x}{\partial s} = 8t $$, $$ \frac{\partial z}{\partial y} = 5e^{3x+5y} $$, $$ \frac{\partial y}{\partial s} = 6 $$

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