 So another partial derivative identity that comes in very handy is the reciprocal rule. And that one is fairly simple. If we have a partial derivative, dx dy at constant z, that's related to, in fact, it's 1 over. It's the reciprocal of the upside down of that partial derivative. So that partial derivative dx dy with respect to dx dy at constant z, that's just 1 over dy dx at constant z. So when I take the reciprocal, I just exchange the numerator and the denominator. Another way of writing that identity, dx dy at constant z, if I move this derivative over to the other side out of the denominator up into the numerator, it looks like dy dx at constant z over here. So if you prefer, you can remember the reciprocal rule as this partial derivative multiplied by its reciprocal gives us 1. So that's a fairly elementary, straightforward partial derivative identity. The question is how to use it. So maybe it'll be more clear if we use a concrete example with some actual numbers. So you're familiar with the heat capacity of a substance like water, so the heat capacity of, let's say, liquid water. In non-SI units, that's exactly 1 in units of calories per gram Kelvin, or in SI units that works out to be about 75 Joules per mole Kelvin. So what that number literally means, because we know that the heat capacity is, by definition, the rate of change of the enthalpy as I change the temperature at constant pressure, that's the constant pressure heat capacity. So this is telling us dH dt at constant P is equal to this number. What that means is in order to raise the temperature by 1 Kelvin, it costs me 75 Joules per mole in order to do that. Sometimes though, we're not interested in how much energy it takes to raise the temperature by a certain amount. Sometimes we provide a certain amount of temperature to a material and we want to know the opposite. How much did it raise the temperature? So if we invert this quantity, so if I'm instead curious about how quickly is the temperature changing as I change the enthalpy. If I supply some heat to a material at a constant pressure, if I change the enthalpy of the material by a certain amount, how much is the temperature going to change? That's just the reciprocal of this quantity. So 1 over 75.4, that's 0.0133 in units of Kelvin moles per Joule, or Kelvin per Joule per mole, if we prefer. So by taking the reciprocal of that quantity, now we have an answer to questions like if I supply 1 Joule per mole of enthalpy to the material, how much does it raise the temperature? It raises the temperature by 0.01 Kelvin or so. So again, fairly straightforward, turn the derivative upside down and it's the value of it is the reciprocal. We can use that to derive relationships between thermodynamic variables. We have seen already that if what we're interested in is how much the Gibbs free energy is changing as we change the volume, the chain rule has already told us that that is equal to volume times dp dv at constant T. dp dv at constant T looks a little bit familiar. That's related to, so as we make this next transformation, dp dv at constant T is not something we have a name for, but the reciprocal of it dv dp at constant T. In particular, if I take 1 over v dv dp at constant T with a negative sign, that's the thing that we've called kappa, the isothermal compressibility. So if I just rearrange that equation a little bit, minus v times kappa is equal to dv dp at constant T. So like I said, this derivative looks a little bit familiar. It's just the reciprocal upside down of something that we have a little bit of experience with. So I can now rewrite this expression and say dg dv at constant T is equal to volume times dp dv at constant T, but that's the reciprocal of dv dp at constant T. So the reciprocal of minus v times kappa. The v's now cancel, and I can say dg dv at constant T is equal to minus 1 over kappa minus 1 over the isothermal compressibility. So now with the combination of the chain rule and the reciprocal rule, we've managed to derive something that's actually fairly useful. If we do want to know dg dv at constant T, that's going to be minus 1 over kappa, and kappa, the isothermal compressibility, is something that we can look up. It's tabulated from many materials. That's a property that's been measured. It's a named property that's been measured for many different substances. So this is a useful relationship that we can actually use to learn something. So there's a couple of other partial derivative identities that are equally useful, and we'll talk about those next.