We are given:

$z = e^{9x + 4y}$

where $x = 4st$ and $y = 3s + 5t$.

To find $z_s$ (the partial derivative of $z$ with respect to $s$):

First, apply the chain rule to $z$ in terms of $s$:

1. Start by differentiating $z$ with respect to $x$ and $y$:

$\frac{\partial z}{\partial x} = e^{9x + 4y} \cdot 9 \quad \text{and} \quad \frac{\partial z}{\partial y} = e^{9x + 4y} \cdot 4$

2. Next, calculate the partial derivatives of $x$ and $y$ with respect to $s$:

$\frac{\partial x}{\partial s} = 4t \quad \text{and} \quad \frac{\partial y}{\partial s} = 3$

3. Now, apply the chain rule to find $z_s$:

$z_s = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial s}$

Substitute the expressions:

$z_s = \left( e^{9x + 4y} \cdot 9 \right) \cdot 4t + \left( e^{9x + 4y} \cdot 4 \right) \cdot 3$

Simplify:

$z_s = e^{9x + 4y} \left( 36t + 12 \right)$

Now substitute the expressions for $x$ and $y$ in terms of $s$ and $t$:

$z_s = e^{9(4st) + 4(3s + 5t)} \left( 36t + 12 \right)$

Simplify further:

$z_s = e^{36st + 12s + 20t} \left( 36t + 12 \right)$

Final answer for $z_s$:

$z_s = e^{36st + 12s + 20t} (36t + 12)$

Would you like me to also find $z_t$, or would you like further clarification on any of the steps?