# Help with simple derivation of probability of credit default

I'm going over a chapter in Hull's Options, Futures, and Other Derivatives and am stuck on how the probability of default is derived. Here's the image of the derivation. I can follow all of it except for one step: how do you derive $$V(t) = e^{-\int_0^t \lambda(\tau) \,d\tau}$$ from $$\frac{dV(t)}{dt} = -\lambda (t)V(t)$$ ?

I'm not a quant so I don't really know how to proceed. I can just plug in the formula in my project, but I'd rather understand how/why the derivation works.

First and foremost, assume that the company can not default at time $$t=0$$, implying that $$V(0)=1$$.
Now, divide with $$V(t)$$ on both sides and integrate from 0 to $$t$$:
$$\int_0^t \frac{\frac{dV(t)}{dt}}{V(t)} dt = - \int_0^t \lambda(t) dt$$ Calculate the LHS: \begin{align} \ln(V(t)) - \ln(V(0)) &= - \int_0^t \lambda(t) dt\\ &\Updownarrow\\ \ln(V(t)) &= - \int_0^t \lambda(t) dt\\ &\Updownarrow\\ V(t) &= e^{ - \int_0^t \lambda(t) dt}, \end{align} where you can substitute $$t$$ with $$\tau$$ in the integrand on the RHS (the hazard rate function) in order to alleviate notational confusion. This will give you the desired result.