Question

Draw a tree diagram of the partial derivatives of the function. The functions are\(t = f(u,v,w)\), where\(u = u(p,q,r,s),v = v(p,q,r,s),w = w(p,q,r,s).\)

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\(t = f(u,v,w)\), where \(u = u(p,q,r,s),v = v(p,q,r,s),w = w(p,q,r,s).\)

Differentiate \(t\) partially with respect to \(p\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial p}} = \frac{\partial }{{\partial p}}f(u,v,w)\\ = \frac{{\partial f}}{{\partial u}} \cdot \frac{{\partial u}}{{\partial p}} + \frac{{\partial f}}{{\partial v}} \cdot \frac{{\partial v}}{{\partial p}} + \frac{{\partial f}}{{\partial w}} \cdot \frac{{\partial w}}{{\partial p}}\\ = {f_u} \cdot {u_p} + {f_v} \cdot {v_p} + {f_w} \cdot {w_p}\end{aligned}\)

Differentiate \(t\) partially with respect to \(q\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial q}} = \frac{\partial }{{\partial q}}f(u,v,w)\\ = \frac{{\partial f}}{{\partial u}} \cdot \frac{{\partial u}}{{\partial q}} + \frac{{\partial f}}{{\partial v}} \cdot \frac{{\partial v}}{{\partial q}} + \frac{{\partial f}}{{\partial w}} \cdot \frac{{\partial w}}{{\partial q}}\\ = {f_u} \cdot {u_q} + {f_v} \cdot {v_q} + {f_w} \cdot {w_q}\end{aligned}\)

Differentiate \(t\) partially with respect to \(r\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial r}} = \frac{\partial }{{\partial r}}f(u,v,w)\\ = \frac{{\partial f}}{{\partial u}} \cdot \frac{{\partial u}}{{\partial r}} + \frac{{\partial f}}{{\partial v}} \cdot \frac{{\partial v}}{{\partial r}} + \frac{{\partial f}}{{\partial w}} \cdot \frac{{\partial w}}{{\partial r}}\\ = {f_u} \cdot {u_r} + {f_v} \cdot {v_r} + {f_w} \cdot {w_r}\end{aligned}\)

Differentiate \(t\) partially with respect to \(s\) as follows.

\(\begin{aligned}{l}\frac{{\partial R}}{{\partial s}} = \frac{\partial }{{\partial s}}f(u,v,w)\\ = \frac{{\partial f}}{{\partial u}} \cdot \frac{{\partial u}}{{\partial s}} + \frac{{\partial f}}{{\partial v}} \cdot \frac{{\partial v}}{{\partial s}} + \frac{{\partial f}}{{\partial w}} \cdot \frac{{\partial w}}{{\partial s}}\\ = {f_u} \cdot {u_s} + {f_v} \cdot {v_s} + {f_w} \cdot {w_s}\end{aligned}\)

Obtain the tree diagram as shown in Figure 1.

In Figure 1, there are two layers, one with partial derivatives of t with respect to \(u,v,w\) and another with partial derivatives of \(u,v,w\)with respect to \(p,q,r,s\) respectively.