Q. 5. Explain why the chain rule from ... [FREE SOLUTION]

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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

$\frac{鈭� z}{鈭� t_{1}} = \frac{鈭� z}{鈭� x_{1}} \frac{鈭� x_{1}}{鈭� t_{1}}$

(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 $z = f (x_{1}, x_{2}, 鈥�, x_{n})$ and $x_{i} = u_{i} (t_{1}, t_{2}, 鈥�, t_{m})$ for $1 鈮� i 鈮� n$ for all values

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

$\frac{鈭� z}{鈭� t_{j}} = \frac{鈭� z}{鈭� x_{1}} \frac{鈭� x_{1}}{鈭� t_{j}} + \frac{鈭� z}{鈭� x_{2}} \frac{鈭� x_{2}}{鈭� t_{j}} + 鈥� + \frac{鈭� z}{鈭� x_{n}} \frac{鈭� x_{n}}{鈭� t_{j}} 鈥� 鈥� (1)$

Where $1 鈮� j 鈮� m$

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 $1 鈮� i$ 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 $(1)$

$\frac{鈭� z}{鈭� t_{1}} = \frac{鈭� z}{鈭� x_{1}} \frac{鈭� x_{1}}{鈭� t_{1}}$

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91影视

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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Most popular questions from this chapter

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91影视

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

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Pure Maths

Discrete Mathematics

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 $m = 1$