Help with simplifying derivatives when sketching graphs

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Homework Help Overview

The discussion revolves around sketching the graph of the function x^(4/9) * e^(-x) and involves simplifying derivatives, particularly the second derivative y''. Participants are exploring the differentiation process and the implications for graphing the function.

Discussion Character

  • Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss different forms of the first derivative y' and their usefulness for further differentiation. There is an emphasis on simplifying y'' and concerns about the length and complexity of the derivative expressions. Questions arise regarding potential errors in differentiation and the process of factoring a quadratic expression.

Discussion Status

Some participants have provided guidance on how to approach the differentiation of y' and suggested starting with a specific form to simplify y''. There is acknowledgment of a potential mistake in the original derivative, and participants are actively engaged in refining their expressions and checking for errors.

Contextual Notes

Participants express difficulty in managing the complexity of derivatives and factoring quadratics, particularly when fractions are involved. There is no consensus on the final form of y'' or the method for factoring, indicating ongoing exploration of these concepts.

TsAmE
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Homework Statement



Sketch the graph of x ^ (4/9) * e ^ (-x)

Homework Equations



None.

The Attempt at a Solution



My y' = -x ^ (4/9) * e ^ (-x) ( 1 - 4/9x ^ 1/9). I keep on getting a reaaally long derivative for y'' and thus cannot place it on my sign table. Could someone please show me the correct steps in order to get a simplified y''?
 
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TsAmE said:

Homework Statement



Sketch the graph of x ^ (4/9) * e ^ (-x)

Homework Equations



None.

The Attempt at a Solution



My y' = -x ^ (4/9) * e ^ (-x) ( 1 - 4/9x ^ 1/9). I keep on getting a reaaally long derivative for y'' and thus cannot place it on my sign table. Could someone please show me the correct steps in order to get a simplified y''?

You can write y' in two ways - as a sum from the product rule, or as a product. Each form is useful for some purpose.

If y = x4/9e-x,
y' = (4/9)x-5/9 e-x - x4/9 e -x ;; as a sum (actually a difference) straight from the product rule
= e-x x-5/9 (4/9 - x) ;; in factored form

The first form is probably easier to differentiate so that you can get y''. The factored form is more useful if you want to find critical numbers and intervals where y' > 0 or y' < 0.

I think your derivative has an error in it.
 
Oh didnt see that mistake thanks. The only problem I am having is finding y'' since you have to differentiate the sum of 2 products or 3 products which has led me to getting y'' to be almost 2 lines long.
 
If you start with the first form I showed above (the difference, not the factored form), you should be able to write y'' with at most four terms.
 
Oh ok I did as you said now I got a nice simplified

y'' = x^(-14/9) * e^(-x) * ( x^(2) - 8/9x - 20/81 )

but I have no idea how I could factorise the quadratic due to the fractions? Unless I have made some mistake, although I doubt it, cause I checked rigorously
 
Your expression for y'' look fine to me. To factor the quadratic, it's probably most efficient to use the quadratic formula.
 

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