Question

Find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=x^{2}+y^{2}+z^{2}, \quad x=t \sin s, \quad y=t \cos s, \quad z=s t^{2}\)

Short answer

The partial derivative of \(w\) with respect to \(s\) equals to \(2x t \cos s - 2y t \sin s + 2z t^{2}\), and the partial derivative of \(w\) with respect to \(t\) is \(2x \sin s + 2y \cos s + 4zt\)

Step-by-step solution

Find the partial derivatives of w

Find the partial derivatives of \(w = x^{2}+y^{2}+z^{2}\) in terms of \(x\), \(y\), and \(z\). \(\partial w / \partial x = 2x\), \(\partial w / \partial y = 2y\), \(\partial w / \partial z = 2z\)

Find the partial derivatives of x, y, and z

Next, derive \(x = t \sin s\), \(y = t \cos s\), and \(z = s t^{2}\) with respect to \(s\) and \(t\). We get that: \(\partial x / \partial s = t \cos s, \partial x / \partial t = \sin s, \partial y / \partial s = -t \sin s, \partial y / \partial t = \cos s, \partial z / \partial s = t^{2}, \partial z / \partial t = 2st\)

Apply the Chain Rule for \(\partial w/ \partial s\)

Applying the Chain Rule, \(\partial w / \partial s\) equals to: \(\partial w / \partial s = (\partial w / \partial x)(\partial x / \partial s) + (\partial w / \partial y)(\partial y / \partial s) + (\partial w / \partial z)(\partial z / \partial s) = 2x t \cos s - 2y t sin s + 2z t^{2}\)

Apply the Chain Rule for \(\partial w/ \partial t\)

Similarly, apply the Chain Rule to find out \(\partial w / \partial t\), which equals to: \(\partial w / \partial t = (\partial w / \partial x)(\partial x / \partial t) + (\partial w / \partial y)(\partial y / \partial t) + (\partial w / \partial z)(\partial z / \partial t) = 2x \sin s + 2y \cos s + 4zt\)