# Differentiating a Payoff

Okay this is probably going to be an extremely easy/straightforward question but I thought I should post it here just to double check. Suppose I have a payoff $\Phi = (S_{T}-K)^{+}$. Now let's say I now have an equation: $u = s\partial_{s}\Phi - \Phi$, this means that given a payoff $\Phi$ as given above then, substituting this payoff into the equation and assuming $S_T = S_{0}\exp((r-1/2)T+\sigma\sqrt{T}Z_i))$ then I should get:

$u = max(S_{T},0) - max(S_{T}-K,0)$, right?

And from this equation, the possible solutions should be:

If $S_{T} > K$, $u = K$, if $S_{T} < K$ and $S_{T} > 0$, $u = S_{T}$, and if $S_{T} < K$ and $S_{T} < 0$ then $u = 0$.

Is all of this correct? I know this is really trivial but I just thought I should check...

$$s\partial_{s}\Phi = S_T I_{S_T>K}.$$ so no.