Find the following derivatives. Type answers in using s and t as the variables
z Subscript s and z Subscript t​, where zequalse Superscript 5 x plus y​, xequals6 st​, and yequals7 s plus 6 t

We are given:

\[
z = e^{5x + y}
\]
\[
x = 6st
\]
\[
y = 7s + 6t
\]

We need to find \( z_s \) (the partial derivative of \( z \) with respect to \( s \)) and \( z_t \) (the partial derivative of \( z \) with respect to \( t \)).

### Step 1: Compute \( z_s \)

By the chain rule, we have:

\[
z_s = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}
\]

- First, calculate \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \):

\[
\frac{\partial z}{\partial x} = 5e^{5x + y}, \quad \frac{\partial z}{\partial y} = e^{5x + y}
\]

- Next, compute \( \frac{\partial x}{\partial s} \) and \( \frac{\partial y}{\partial s} \):

\[
\frac{\partial x}{\partial s} = 6t, \quad \frac{\partial y}{\partial s} = 7
\]

Now, substitute everything into the equation for \( z_s \):

\[
z_s = 5e^{5x + y} \cdot 6t + e^{5x + y} \cdot 7
\]

Factor out \( e^{5x + y} \):

\[
z_s = e^{5x + y}(30t + 7)
\]

### Step 2: Compute \( z_t \)

Similarly, by the chain rule:

\[
z_t = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t}
\]

- We already know \( \frac{\partial z}{\partial x} = 5e^{5x + y} \) and \( \frac{\partial z}{\partial y} = e^{5x + y} \).

- Now compute \( \frac{\partial x}{\partial t} \) and \( \frac{\partial y}{\partial t} \):

\[
\frac{\partial x}{\partial t} = 6s, \quad \frac{\partial y}{\partial t} = 6
\]

Substitute everything into the equation for \( z_t \):

\[
z_t = 5e^{5x + y} \cdot 6s + e^{5x + y} \cdot 6
\]

Factor out \( e^{5x + y} \):

\[
z_t = e^{5x + y}(30s + 6)
\]

### Final Answers:

\[
z_s = e^{5x + y}(30t + 7)
\]
\[
z_t = e^{5x + y}(30s + 6)
\]

Would you like a more detailed explanation or have any questions?

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**Relative Questions:**
1. How does the chain rule apply when differentiating functions of multiple variables?
2. Can you provide a general form for partial derivatives using the chain rule?
3. What is the physical interpretation of partial derivatives in a multivariable context?
4. How do exponential functions behave when differentiated?
5. How do product and chain rules combine in more complex derivatives?

**Tip:** When differentiating, always look for opportunities to factor common terms for simpler results.

Find the following derivatives. Type answers in using s and t as the variables: z_subscript_s and z_subscript_t, where z = e^(5x + y), x = 6st, and y = 7s + 6t.

This problem involves finding the partial derivatives z_s and z_t for the function z = e^(5x + y), where x = 6st and y = 7s + 6t. It uses the chain rule for multivariable calculus, demonstrating how to compute derivatives for exponential functions with respect to multiple variables. Ideal for university-level students studying advanced calculus.

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