# 2.2: Partial Derivatives

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The equation

\[ z = z (x,~ y)\]

represents a two-dimensional surface in three-dimensional space. The surface intersects the plane y = constant in a plane curve in which *z* is a function of *x*. One can then easily imagine calculating the slope or gradient of this curve in the plane *y* = constant. This slope is \( \left( \frac{\partial z}{\partial x} \right)_y\) - the partial derivative of z with respect to x, with y being held constant. For example, if

\[ z = y \ln x,\]

then

\[ \left( \frac{\partial z}{\partial x} \right)_y = \frac{y}{x},\]

*y* being treated as though it were a constant, which, in the plane *y* = constant, it is. In a similar manner the partial derivative of *z* with respect to *y*, with *x* being held constant, is

\[ \left( \frac{\partial z}{\partial y} \right)_x = \ln x\]

When you have only three variables – as in this example – it is usually obvious which of them is being held constant. Thus *∂z/∂y* can hardly mean anything other than at constant *x*. For that reason, the subscript is often omitted. In thermodynamics, there are often more than three variables, and it is usually (I would say always) essential to indicate by a subscript which quantities are being held constant.

In the matter of pronunciation, various attempts are sometimes made to give a special pronunciation to the symbol ∂. (I have heard “day”, and “dye”.) My own preference is just to say “partial dz by dy”.

Let us suppose that we have evaluated z at *(x , y)*. Now if you increase *x* by δx, what will the resulting increase in *z* be? Obviously, to first order, it is \( \frac{\partial x}{\partial x} \delta x\). And if *y* increases by δ*y*, the increase in *z* will be \( \frac{\partial z}{\partial y} \delta y\). And if both *x* and *y* increase, the corresponding increase in *z*, to first order, will be

\[ \delta z = \frac{\partial z}{\partial x} \delta x + \frac{\partial z}{\partial y} \delta y\]

No great and difficult mathematical proof is needed to “derive” this; it is just a plain English statement of an obvious truism. The increase in *z* is equal to the rate of increase of *z* with respect to *x* times the increase in *x* plus the rate of increase of *z* with respect to *y* times the increase in *y*.

Likewise if *x* and *y* are increasing with time at rates \( \frac{dx}{dt}\) and \( \frac{dy}{dt}\), the rate of increase of *z* with respect to time is

\[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}.\]