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Integrals: I can do but I am confused about the limit

Derivative: I am having problem. Need some help with that. I have a solution but I think it is wrong.

For each problem I have posted step by step what I am doing. Hopefully it will be easier in answering.

__Problem 1: Integral__What is [tex]\int e ^{-t}[/tex] (limit is from 0 (on bottom) to 1 (on top))

Problem 1: My solutionProblem 1: My solution

My issue is step 6

1. Take u = -t

2. Use Chain Rule: dy/dx = (dy/du)(du/dx)

3. du/dt = -1. Therefor -du = dt

4. Find limits:

__For t=1:__u= -1

__For t=0:__u= 0

5. New integral becomes (-1) [tex]\int e ^{u} du [/tex].

6. My question/issue is about the new limits:

Do I have to evaluate the integral with u=-1 on bottom of integral and u=0 on top of integral or the other way.

__Problem 2 Statement:__y= [tex]x^{x^x}[/tex]. Find the minimum of y.

Problem 2 Solution

I am not really sure that I am doing this right. So plz advise

I know how to find the derivative of [tex] x^x [/tex].

So here are my steps

1. make a = [tex] x^x [/tex] (I am taking the first two x's)

2. Therefore y = [tex] a^x [/tex]

3. dy/dx = (dy/da)(da/dx)

4. Find dy/da:

y = [tex] a^x [/tex]

= [tex] e^{xlna} [/tex]

dy/da = [tex] a^x [/tex] d(xlna)/da = [tex] (a^x) [/tex] [tex] x^{1-x} [/tex]

5. Find da/dxa = [tex] x^x [/tex] = [tex] e^{xlnx} [/tex]

By productrule da/dx = [tex] x^x[/tex] (ln x + 1)

6. Multiple from step 4,5 and equate to 0.

My question is: is step 4 right? Actually now that I am looking at it, I am wondering if step 5 is right also. Plz advise on this.

Thanks

Asif