Help with simplifying derivatives when sketching graphs

[TsAmE]

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''?

 

[Mark44]

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 = x^4/9e^-x,
y' = (4/9)x^-5/9 e^-x - x^4/9 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.

 

[TsAmE]

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.

 

[Mark44]

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.

 

[TsAmE]

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

 

[Mark44]

Your expression for y'' look fine to me. To factor the quadratic, it's probably most efficient to use the quadratic formula.