Question

Draw a tree diagram of the partial derivatives of the function. The functions are\({\rm{R = f(x, y, z, t)}}\), where\({\rm{x = x(u, v, w), y = y(u, v, w), z = z(u, v, w)}}\), and \({\rm{t = t(u, v, w)}}{\rm{.}}\)

Short answer

The answer is stated below.

Step-by-step solution

The chain rule.

With the chain rule, the partial derivative of function\({\rm{f(x, y)}}\)with respect to\({\rm{t}}\)becomes\(\frac{{\partial f}}{{\partial t}} = \frac{{\partial f}}{{\partial x}} \cdot \frac{{\partial x}}{{\partial t}} + \frac{{\partial f}}{{\partial y}} \cdot \frac{{\partial y}}{{\partial t}}.\)

Use the chain rule for calculation and obtain the diagram.

The functions are\(R = f(x,y,z,t)\), where \(x = x(u,v,w),y = y(u,v,w){\rm{,}}z = z(u,v,w)\) and \(t = t(u,v,w){\rm{.}}\)

Differentiate \(R\) partially with respect to \(u\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial u}} = \frac{\partial }{{\partial u}}f(x,y,z,t)\\ = \frac{{\partial f}}{{\partial x}} \cdot \frac{{\partial x}}{{\partial u}} + \frac{{\partial f}}{{\partial y}} \cdot \frac{{\partial y}}{{\partial u}} + \frac{{\partial f}}{{\partial z}} \cdot \frac{{\partial z}}{{\partial u}} + \frac{{\partial f}}{{\partial t}} \cdot \frac{{\partial t}}{{\partial u}}\\ = {f_x} \cdot {x_u} + {f_y} \cdot {y_u} + {f_z} \cdot {z_u} + {f_t} \cdot {t_u}\end{aligned}\)

Differentiate \(R\) partially with respect to \(v\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial v}} = \frac{\partial }{{\partial v}}f(x,y,z,t)\\ = \frac{{\partial f}}{{\partial x}} \cdot \frac{{\partial x}}{{\partial v}} + \frac{{\partial f}}{{\partial y}} \cdot \frac{{\partial y}}{{\partial v}} + \frac{{\partial f}}{{\partial z}} \cdot \frac{{\partial z}}{{\partial v}} + \frac{{\partial f}}{{\partial t}} \cdot \frac{{\partial t}}{{\partial v}}\\ = {f_x} \cdot {x_v} + {f_y} \cdot {y_v} + {f_z} \cdot {z_v} + {f_t} \cdot {t_v}\end{aligned}\)

Differentiate \(R\) partially with respect to \(w\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial w}} = \frac{\partial }{{\partial w}}f(x,y,z,t)\\ = \frac{{\partial f}}{{\partial x}} \cdot \frac{{\partial x}}{{\partial w}} + \frac{{\partial f}}{{\partial y}} \cdot \frac{{\partial y}}{{\partial w}} + \frac{{\partial f}}{{\partial z}} \cdot \frac{{\partial z}}{{\partial w}} + \frac{{\partial f}}{{\partial t}} \cdot \frac{{\partial t}}{{\partial w}}\\ = {f_x} \cdot {x_w} + {f_y} \cdot {y_w} + {f_z} \cdot {z_w} + {f_t} \cdot {t_w}\end{aligned}\)

Obtain the tree diagram as shown in Figure 1.

In Figure 1, there are two layers, one with partial derivatives of \(R\) with respect to \(x,y,z,t\) and another with partial derivatives of \(x,y,z,t\)with respect to \(u,v,w\) respectively.