Instantaneous forward rate ($f(t,T)=-{\partial }_T\log P(t,T)$)

The continuously compounded rate inferred for an infinitesimal investment interval at future maturity $T$, as seen at time $t$. For a unit-face-value zero-coupon bond, $P(t,T)=\exp (-{\int }_t^{T}f(t,u)du)$ and the short rate is $f(t,t)$.