Final answer:
Using the chain rule for multivariable functions, the partial derivative \(\frac{\partial w}{\partial t}\) for w=f(x,y,z) is \(\frac{\partial w}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial t}\), neglecting \(\frac{\partial x}{\partial t}\) since x does not depend on t.
Step-by-step explanation:
To find the partial derivative \(\frac{\partial w}{\partial t}\) for w=f(x,y,z) where x=g(r,s), y=h(t), and z=k(r,s,t), we need to employ the chain rule for partial derivatives.
According to the chain rule for multivariable functions, the derivative of w with respect to t can be expressed as follows:
\(\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial t}\)
But since x is not a function of t, its derivative with respect to t is zero, simplifying our formula:
\(\frac{\partial w}{\partial t} = \frac{\partial w}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial t}\)
In this formula, y and z are both functions of t, and we must compute their derivatives with respect to t separately:
\(\frac{\partial y}{\partial t}\) and \(\frac{\partial z}{\partial t}\)
After finding these, we can substitute them back into our chain rule expression to find the desired partial derivative of w with respect to t.