To compute the partial derivative of $w$ with respect to $s$, we first note that the given function is:

$w = -3st(e^{st}) - 3t^2(e^{st}) - 5s(t^3)$

Now, we'll take the partial derivative of each term with respect to $s$, treating $t$ as a constant.

First term: $-3st(e^{st})$

Apply the product rule to $-3st(e^{st})$, where:

• The first factor is $-3st$,
• The second factor is $e^{st}$.

The product rule is: $\frac{\partial}{\partial s} [uv] = u \frac{\partial v}{\partial s} + v \frac{\partial u}{\partial s}$

So: $\frac{\partial}{\partial s} [-3st(e^{st})] = (-3st) \cdot \frac{\partial}{\partial s}(e^{st}) + (e^{st}) \cdot \frac{\partial}{\partial s} (-3st)$

1. $\frac{\partial}{\partial s}(e^{st}) = t e^{st}$,
2. $\frac{\partial}{\partial s} (-3st) = -3t$.

Thus: $\frac{\partial}{\partial s} [-3st(e^{st})] = -3st \cdot t e^{st} + e^{st} \cdot (-3t) = -3t^2s e^{st} - 3t e^{st}$

Second term: $-3t^2(e^{st})$

Since $-3t^2$ is constant with respect to $s$, we just differentiate $e^{st}$ with respect to $s$: $\frac{\partial}{\partial s} [-3t^2(e^{st})] = -3t^2 \cdot t e^{st} = -3t^3 e^{st}$

Third term: $-5s(t^3)$

Here, $-5(t^3)$ is constant with respect to $s$, so the derivative is: $\frac{\partial}{\partial s} [-5s(t^3)] = -5(t^3)$

Putting it all together:

The partial derivative of $w$ with respect to $s$ is: $\frac{\partial w}{\partial s} = -3t^2s e^{st} - 3t e^{st} - 3t^3 e^{st} - 5t^3$

Simplifying: $\frac{\partial w}{\partial s} = -3t^2s e^{st} - 3(t^3 + t) e^{st} - 5t^3$

This is the partial derivative of $w$ with respect to $s$.

Would you like a more detailed explanation of any part? Here are some related questions you might find useful:

1. How does the product rule apply to more complex functions?
2. What happens if both variables are considered in the derivative?
3. What is the general rule for differentiating exponential functions with variable exponents?
4. How do constants affect the derivative in multivariable calculus?
5. Could this result change if the roles of $s$ and $t$ were switched?

Tip: When differentiating exponentials like $e^{st}$, always remember to apply the chain rule for variable exponents!