Proof of product rule for differentiation using chain rule for partial differentiation

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This article proves the product rule for differentiation in terms of the chain rule for partial differentiation.

Contents

1 Statements
- 1.1 Statement of product rule for differentiation (that we want to prove)
- 1.2 Statement of chain rule for partial differentiation (that we want to use)
2 Proof

Statements

Statement of product rule for differentiation (that we want to prove)

uppose $\text{[math]}$ and $\text{[math]}$ are functions of one variable. Then the following is true wherever the right side expression makes sense (see concept of equality conditional to existence of one side):

$\text{[math]}$

Statement of chain rule for partial differentiation (that we want to use)

Suppose $\text{[math]}$ are both functions of one variable and $\text{[math]}$ is a function of two variables. Suppose $\text{[math]}$ . Then:
$\text{[math]}$

Proof

Given: Functions $\text{[math]}$ and $\text{[math]}$

To prove: $\text{[math]}$ wherever the right side makes sense.

Proof:

Consider the function:

$\text{[math]}$

Its partial derivatives are:

$\text{[math]}$ Define:

$\text{[math]}$

By the chain rule for partial differentiation, we have:

$\text{[math]}$

The left side is $\text{[math]}$ . The right side becomes:

$\text{[math]}$

This simplifies to:

$\text{[math]}$

Plug back the expressions $\text{[math]}$ and get:

$\text{[math]}$

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