Understanding the Product Rule for Differential Operators

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Homework Help Overview

The discussion revolves around the application of the product rule in the context of differential operators, specifically focusing on the expression involving the partial derivatives with respect to variables defined in terms of other variables.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to understand how the product rule applies to the given expressions for h and g. Questions are raised about the meaning of the differential operator ∂/∂x and its relationship to the product rule.

Discussion Status

Some participants express familiarity with the basic definitions of partial derivatives but indicate uncertainty regarding the specific application of the product rule to arrive at the stated result. There is a recognition that the result may not directly follow from the product rule as initially thought.

Contextual Notes

Participants are grappling with the definitions and implications of differential operators in the context of the problem, indicating a need for further clarification on the assumptions underlying the product rule's application.

zezima1
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From my book:

Let

h = x + ay and g = x + by

By product rule we then have:

∂/∂x = ∂/∂h + ∂/∂g

Can someone explain to me how to arrive at this result? I don't even get what ∂/∂x even means, isn't it just a differential operator?
 
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zezima1 said:
From my book:

Let

h = x + ay and g = x + by

By product rule we then have:

∂/∂x = ∂/∂h + ∂/∂g

Can someone explain to me how to arrive at this result? I don't even get what ∂/∂x even means, isn't it just a differential operator?
Yes, ∂/∂x is an operator.

Do you know the definition of [itex]\displaystyle \frac{\partial F(x,\,y)}{\partial x}[/itex], and/or how to evaluate this partial derivative given the function, F(x, y) ?
 
yes yes I know all that, I just can't specifically see how they arrive at the result using the product rule :)
 
zezima1 said:
yes yes I know all that, I just can't specifically see how they arrive at the result using the product rule :)
Off hand, I don't see the result as coming from the product rule either.
 

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