Given the function $z = x^4 y^2 - y^4 x^3$ and the relations $x = s + t$ and $y = \frac{s}{t}$, we need to find the partial derivative $\frac{\partial z}{\partial t}$ when $(t, s) = (1, 2)$.

Step 1: Express $z$ in terms of $t$ and $s$

We first rewrite the expressions for $x$ and $y$ in terms of $s$ and $t$:

• $x = s + t$
• $y = \frac{s}{t}$

Substitute these into the equation for $z$:

$z = (s + t)^4 \left( \frac{s}{t} \right)^2 - \left( \frac{s}{t} \right)^4 (s + t)^3$

Step 2: Differentiate $z$ with respect to $t$

To find $\frac{\partial z}{\partial t}$, apply the product rule and chain rule.

We now differentiate both terms of $z$ with respect to $t$, considering that $x$ and $y$ are functions of $t$.

Term 1: $(s + t)^4 \left( \frac{s}{t} \right)^2$

Differentiate with respect to $t$ using the product rule.

Term 2: $\left( \frac{s}{t} \right)^4 (s + t)^3$

Differentiate with respect to $t$ using the product rule.

Let me now compute these derivatives and evaluate them at $t = 1$ and $s = 2$.The value of $\frac{\partial z}{\partial t}$ when $(t, s) = (1, 2)$ is 1080.

Would you like more details on how the derivative was computed, or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How do you compute partial derivatives for multivariable functions in general?
2. What is the significance of the product and chain rules in multivariable calculus?
3. How would the derivative change if the function $z$ had more complex dependencies on $x$ and $y$?
4. What if $x$ and $y$ were functions of other variables, such as $r$ and $\theta$?
5. Can you explain how the quotient rule would apply to the differentiation of the terms involving $y = \frac{s}{t}$?

Tip: Always simplify expressions involving variables before taking derivatives—it helps avoid complex terms during differentiation.