More formal treatment of multivariable chain rule | Summary and Q&A

TL;DR

This video explains the formal argument for the multivariable chain rule, which involves understanding the vector value derivative and the directional derivative.

Key Insights

• 📏 The multivariable chain rule involves understanding the vector value derivative and the directional derivative.
• 😑 The formal argument begins with the definition of a derivative and manipulates the expressions to represent the nudge in the input and the resulting change in the intermediary space.
• 👾 The vector value derivative is crucial in understanding the multivariable chain rule as it represents the change in the intermediary space due to a nudge in the input.

Transcript

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Questions & Answers

Q: What does the multivariable chain rule involve?

The multivariable chain rule involves a vector-valued function and a function that maps high-dimensional space to a number line.

Q: How is the formal argument for the multivariable chain rule derived?

The argument begins by using the formal definition of a derivative and manipulating the expressions to represent the nudge in the input and the resulting change in the intermediary space.

Q: Why is the vector value derivative important in the multivariable chain rule?

The vector value derivative represents the change in the intermediary space as a result of a nudge in the input. It plays a crucial role in understanding the multivariable chain rule.

Q: What is the role of the directional derivative in the multivariable chain rule?

The directional derivative, calculated using the gradient of the function and the vector value derivative, allows us to determine the rate of change of the composition function in the direction of the derivative of the function.

Summary & Key Takeaways

• The video discusses the multivariable chain rule, which involves a vector-valued function and a function that maps high-dimensional space to a number line.

• The formal argument begins with the definition of a derivative, where h represents dt.

• Nudging the input by dt causes a change in the intermediary space, represented by a vector-value derivative.

• By manipulating the expressions and using the definition of the directional derivative, the multivariable chain rule is derived.