# Thread: Derivative of implicit function (risk & reward)

1. ## Derivative of implicit function (risk & reward)

Hi members,
I want to differentiate w.r.t.$\sigma^2$ the following equation.
$u'(Y)+\frac{u''(Y)}{2}(\sigma^2 + \mu^2) = 0$
where we can consider $\mu$(reward) as an implicit function of $\sigma^2$(risk) of small bets
$u'(Y)\frac{du}{d\sigma^2}+\frac{u''(Y)}{2}+\mu u''(Y)\frac{du}{d\sigma^2}=0$

Hence,
$u'(\sigma^2)=\frac{u''(Y)}{2(u'(Y)+\mu u''(Y)}$
But answer given in the PDF downloaded from internet is $u'(\sigma^2)=\frac{u''(Y)}{2(u'(Y)+\mu(\sigma^2)u ''(Y)}$

What does 1st and 2nd derivative of Utility function imply?

2. ## Re: Derivative of implicit function (risk & reward)

Hey Vinod.

Did you consider the chain rule? Also what is mu a function of specifically?

3. ## Re: Derivative of implicit function (risk & reward)

Originally Posted by chiro
Hey Vinod. Also what is mu a function of specifically?
$\displaystyle \mu$ is the mean representing the expected return (most likely expected value of lognormal distribution)

4. ## Re: Derivative of implicit function (risk & reward)

So is it a constant? The answer seems to suggest that it is at least a function of sigma^2.