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:

y(x)=e^{a|x|}

I'm pretty sure its not as simple as

y'(x)=a e^{a|x|}

Any suggestions??

 

[mrandersdk]

i think the problem is that |x| is not differential in zero so, e^{a|x|} 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 x<0

\partial_x e^{a|x|} = \partial_x e^{-ax} = -ae^{-ax}

for x>0 you get

\partial_x e^{a|x|} = \partial_x e^{ax} = ae^{ax}

combining these could be

\partial_x e^{a|x|} = sign(x) a e^{a|x|} = \frac{x}{|x|} a e^{a|x|}

but remember that it is not defined in 0.

 

[robousy]

Hmmm,

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

Thats cool. :)

Thanks mranderson, very helpful.