Given the function $f(y, z) = (\sin y) e^{z^3 - z - 1} + e^{yz}$, we will compute each of the partial derivatives step by step.

(a) Determine $f_z$ (the partial derivative with respect to $z$):

We apply the rules of differentiation, treating $y$ as a constant.

First term:

$(\sin y) e^{z^3 - z - 1}$

The derivative of $e^{z^3 - z - 1}$ with respect to $z$ is: $\frac{d}{dz}\left(e^{z^3 - z - 1}\right) = e^{z^3 - z - 1} \cdot \frac{d}{dz}(z^3 - z - 1) = e^{z^3 - z - 1} \cdot (3z^2 - 1)$

Thus, the derivative of the first term is: $(\sin y) \cdot e^{z^3 - z - 1} \cdot (3z^2 - 1)$

Second term:

$e^{yz}$

The derivative of $e^{yz}$ with respect to $z$ is: $\frac{d}{dz}(e^{yz}) = y e^{yz}$

So the partial derivative $f_z$ is: $f_z = (\sin y) e^{z^3 - z - 1} (3z^2 - 1) + y e^{yz}$

(b) Determine $f_y$ (the partial derivative with respect to $y$):

We now differentiate with respect to $y$, treating $z$ as constant.

First term:

$(\sin y) e^{z^3 - z - 1}$

The derivative of $\sin y$ is $\cos y$. So, the derivative of the first term is: $(\cos y) e^{z^3 - z - 1}$

Second term:

$e^{yz}$

The derivative of $e^{yz}$ with respect to $y$ is: $\frac{d}{dy}(e^{yz}) = z e^{yz}$

So the partial derivative $f_y$ is: $f_y = (\cos y) e^{z^3 - z - 1} + z e^{yz}$

(c) Determine $f_{yz}$ (the mixed partial derivative with respect to $y$ and $z$):

Now, we differentiate $f_y$ with respect to $z$.

From part (b), we know that: $f_y = (\cos y) e^{z^3 - z - 1} + z e^{yz}$

First term:

$(\cos y) e^{z^3 - z - 1}$

The derivative with respect to $z$ is: $(\cos y) \cdot e^{z^3 - z - 1} \cdot (3z^2 - 1)$

Second term:

$z e^{yz}$

The derivative with respect to $z$ is: $\frac{d}{dz}(z e^{yz}) = e^{yz} + z y e^{yz}$

So the mixed partial derivative $f_{yz}$ is: $f_{yz} = (\cos y) e^{z^3 - z - 1} (3z^2 - 1) + e^{yz} + z y e^{yz}$

Summary of Results:

(a) $f_z = (\sin y) e^{z^3 - z - 1} (3z^2 - 1) + y e^{yz}$

(b) $f_y = (\cos y) e^{z^3 - z - 1} + z e^{yz}$

(c) $f_{yz} = (\cos y) e^{z^3 - z - 1} (3z^2 - 1) + e^{yz} + z y e^{yz}$

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. What is the second partial derivative $f_{zz}$ of the function?
2. How does the chain rule apply when computing partial derivatives?
3. Can you find the critical points of the function $f(y, z)$?
4. How would the function change if the exponent in $e^{yz}$ were $y^2 z^2$?
5. What happens if $y = 0$ or $z = 0$? How does this simplify $f(y, z)$?

Tip: When differentiating exponentials with a variable in the exponent, remember to apply the chain rule carefully!