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### Essential Calculus: Early Transcendentals

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

See the step by step solution

## Step 1: The chain rule.

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

## Step 2: 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.

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