Quicky on derivative of absolute value in exponential

[robousy]

Hey folks,

I'm looking for a little guidance in solving the derivative y'(x)of the following function containing an absolute in the exponent:

[tex]y(x)=e^{a|x|}[/tex]

I'm pretty sure its not as simple as

[tex]y'(x)=a e^{a|x|}[/tex]

Any suggestions??

 

[mrandersdk]

i think the problem is that [tex] |x| [/tex] is not differential in zero so, [tex]e^{a|x|}[/tex] is not differential in zerp, so if you want to calculate the differential somewhere else, then just do in the two cases. Then you get for [tex]x<0[/tex]

[tex]\partial_x e^{a|x|} = \partial_x e^{-ax} = -ae^{-ax} [/tex]

for [tex]x>0[/tex] you get

[tex]\partial_x e^{a|x|} = \partial_x e^{ax} = ae^{ax} [/tex]

combining these could be

[tex]\partial_x e^{a|x|} = sign(x) a e^{a|x|} = \frac{x}{|x|} a e^{a|x|}[/tex]

but remember that it is not defined in 0.

 

[robousy]

Hmmm,

so the [tex]\frac{x}{|x|}[/tex] is really just a neat way of setting the coefficient to [tex]\pm 1[/tex], depending on where x is.

Thats cool. :)

Thanks mranderson, very helpful.