The Mystery of Green-Colored Steps: A Homework Tutorial

AI Thread Summary
The discussion focuses on understanding a specific mathematical step involving the chain rule in calculus, particularly how to express a derivative in terms of differentials. The participant struggles with manipulating the notation and recognizes the importance of the chain rule in this context. They seek clarification on the technique used to achieve the green-colored step in their homework. After revising their equations, they confirm that the step is indeed clearer when viewed through the lens of the chain rule. Overall, the conversation emphasizes the challenges of applying calculus concepts correctly.
athrun200
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Homework Statement



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Homework Equations





The Attempt at a Solution


In part a, there is an important step, I don't know how to have the step that written in green colour.
Does this technque have any name?
Can you give me more example of it?

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You just manipulate the differentials.
\frac{d \dot{r}}{dt}=\frac{d \dot{r}}{dt} \frac{dr}{dr}=\frac{d \dot{r}}{dr} \frac{dr}{dt}=\dot{r} \frac{d \dot{r}}{dt}

edit. corrected last part
 
Last edited:
bp_psy said:
You just manipulate the differentials.
\frac{d \dot{r}}{dt}=\frac{d \dot{r}}{dt} \frac{dr}{dr}=\frac{d \dot{r}}{dr} \frac{dr}{dt}=\dot{r} \frac{dr}{dr}

It is quick difficult to think of this step.
If I don't check the solution, I wouldn't write out this.
 
athrun200 said:
It is quick difficult to think of this step.
If I don't check the solution, I wouldn't write out this.

It appears that I messed up the last equality. I corrected the last post.
It is clearer if you see that it is the chain rule.
let \dot{r}=\dot{r}(r(t)) then \frac{d \dot{r} }{dt}=\frac{d \dot{r} }{dr} \frac{d{r} }{dt}
 
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