few questions on finding the derivative

[Guest]

hello,

I have a few questions:

1. Whats the derivative of e^-x again? is it e^-x(-1)? or e^-x? I know the derivaative for e^x is e^x.

2. Whats the critical numbers for y= x( 1-x^2)^.5
y'=1(1-x^2)^.5+x(.5)(1-x^2)(-2x)
y'=(1-x^2)^.5-x^2(1-x^2)
0=(1-x^2)^.5(1-x^2(1-x^2)^.5

I got stuck here.

3. Is it always true an increasing function is always concave up in shape?


thanks for the help

 

[galactus]

2. Whats the critical numbers for y= x( 1-x^2)^.5
y'=1(1-x^2)^.5+x(.5)(1-x^2)(-2x)
y'=(1-x^2)^.5-x^2(1-x^2)
0=(1-x^2)^.5(1-x^2(1-x^2)^.5

I got stuck here.

Product rule:

\(\displaystyle \L\\x(\frac{-x}{\sqrt{1-x^{2}}})+\sqrt{1-x^{2}}(1)\)

\(\displaystyle \L\\=\frac{-x^{2}}{\sqrt{x^{2}-1}}+\sqrt{x^{2}-1}\)

Multiply by \(\displaystyle \sqrt{1-x^{2}}\)

\(\displaystyle \L\\\sout{\sqrt{1-x^{2}}}\frac{-x^{2}}{\sout{\sqrt{1-x^{2}}}}+\sqrt{1-x^{2}}\sqrt{1-x^{2}}\)

\(\displaystyle \L\\=-x^{2}+(1-x^{2})\)

Now, set to 0 and solve for x.

 

[skeeter]

1. d/dx(e^u) = e^u*(du/dx)

3. no ... what about the following functions that are increasing throughout their domain?

y = ln(x)

y = sqrt(x)

y = x

what is their concavity?