Final answer:
The question requires using the chain rule to find partial derivatives of the function z with respect to variables s and t, with given relationships for x and y. By applying the chain rule, the derivatives are found in terms of x and y, and then x and y are expressed in terms of s and t.
Step-by-step explanation:
The question asks to use the chain rule to find \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\) for the function z = (x - y)^9, where x = s^2t and y = st^2. First, we apply the chain rule to differentiate z with respect to s and t.
For \(\frac{\partial z}{\partial s}\), we get:
\(\frac{\partial z}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial s}\)
\(= 9(x - y)^8 \times 2st + (-9)(x - y)^8 \times t^2\)
\(= 9(x - y)^8(2st - t^2)\)
For \(\frac{\partial z}{\partial t}\), we get:
\(\frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}\)
\(= 9(x - y)^8 \times s^2 + (-9)(x - y)^8 \times 2st\)
\(= 9(x - y)^8(s^2 - 2st)\)
We then substitute the expressions for x and y in terms of s and t into the derivatives to evaluate \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\) specifically for this problem.