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Q14E

Expert-verifiedFound in: Page 656

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**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 \({\rm{t = t(u, v, w)}}{\rm{.}}\)**

The answer is stated below.

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

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.

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