Chapter 12: Q. 5. (page 963)

Explain why the chain rule from Chapter $2$ is a special case
of Theorem $12.34$ with $n = 1$ and $m = 1$

Short Answer

Expert verified

The required answer is

$\frac{\partial z}{\partial t_{1}} = \frac{\partial z}{\partial x_{1}} \frac{\partial x_{1}}{\partial t_{1}}$

(Since the function is of a single variable so partial derivative

is the same as a normal derivative)

Step by step solution

Given information

The complete version of the chain rule is for a given function $z = f (x_{1}, x_{2}, \dots, x_{n})$ and $x_{i} = u_{i} (t_{1}, t_{2}, \dots, t_{m})$ for $1 \leq i \leq n$ for all values

$t_{1}, t_{2}, \dots, t_{m}$ at which each $u_{i}$ is differentiable and if $f$ is differentiable at $(x_{1}, x_{2}, \dots, x_{n})$ then

$\frac{\partial z}{\partial t_{j}} = \frac{\partial z}{\partial x_{1}} \frac{\partial x_{1}}{\partial t_{j}} + \frac{\partial z}{\partial x_{2}} \frac{\partial x_{2}}{\partial t_{j}} + \dots + \frac{\partial z}{\partial x_{n}} \frac{\partial x_{n}}{\partial t_{j}} \dots \dots (1)$

Where $1 \leq j \leq m$

The objective is to show when n=m=1 then the complete version of the chain rule gives the chain rule for a single variable.

When $n = m = 1$ then the complete version of the chain rule is as follows. For a given function $z = f (x_{1})$ and $x_{i} = u_{i} (t_{1})$ for $1 \leq i$ for the values of

$t_{1}$ at which $u_{1}$ is differentiable and if $f$ is differentiable at $x_{1}$

Then put $n = m = 1$ in the equation $(1)$

$\frac{\partial z}{\partial t_{1}} = \frac{\partial z}{\partial x_{1}} \frac{\partial x_{1}}{\partial t_{1}}$

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Explain why the chain rule from Chapter $2$ is a special case
of Theorem $12.34$ with $n = 1$ and $m = 1$

Short Answer

Step by step solution

Given information

The objective is to show when n=m=1 then the complete version of the chain rule gives the chain rule for a single variable.

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Explain why the chain rule from Chapter 2is a special caseof Theorem 12.34 with n=1and m=1

Short Answer

Step by step solution

Given information

The objective is to show when n=m=1 then the complete version of the chain rule gives the chain rule for a single variable.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

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Applied Mathematics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Explain why the chain rule from Chapter $2$ is a special case
of Theorem $12.34$ with $n = 1$ and $m = 1$