# Quicky on derivative of absolute value in exponential

## Main Question or Discussion Point

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??

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.

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. :)