We are asked to find $\frac{\partial w}{\partial t}$ using the chain rule for the given functions in problem 7: $w = x^2 y$, $x = st$, and $y = s - t$. We need to express the final answer in terms of $s$ and $t$.

1. Problem Description

We are asked to find $\frac{\partial w}{\partial t}$ using the chain rule for the given functions in problem 7:

$w = x^2 y$, $x = st$, and $y = s - t$. We need to express the final answer in terms of $s$ and $t$.

2. Solution Steps

We use the chain rule to find $\frac{\partial w}{\partial t}$:

$\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial t}$

First, we compute the partial derivatives:

$\frac{\partial w}{\partial x} = 2xy$

$\frac{\partial w}{\partial y} = x^2$

$\frac{\partial x}{\partial t} = s$

$\frac{\partial y}{\partial t} = -1$

Now, we substitute these into the chain rule formula:

$\frac{\partial w}{\partial t} = (2xy)(s) + (x^2)(-1)$

$\frac{\partial w}{\partial t} = 2xys - x^2$

Next, we substitute $x = st$ and $y = s - t$ into the equation:

$\frac{\partial w}{\partial t} = 2(st)(s - t)s - (st)^2$

$\frac{\partial w}{\partial t} = 2s^2t(s - t) - s^2t^2$

$\frac{\partial w}{\partial t} = 2s^3t - 2s^2t^2 - s^2t^2$

$\frac{\partial w}{\partial t} = 2s^3t - 3s^2t^2$

3. Final Answer

$\frac{\partial w}{\partial t} = 2s^3t - 3s^2t^2$