97.5k views
5 votes
Use the chain rule to find ∂z/∂s and ∂z/∂t for z = (x - y)⁹, x = s²t, y = st²?

1 Answer

2 votes

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.

User Daviesgeek
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories