# Partial derivatives with only one variable held constant

In thermodynamics, partial derivatives state which variable is held constant. For example $$\frac{\partial U}{\partial V}\vert_T$$ means the partial derivative of the internal energy $$U$$ with respect to the volume $$V$$, keeping the temperature $$T$$ constant.

What happens to the other variables in the function $$U$$ if they are not held constant? As an example, what would $$\frac{\partial f}{\partial x}\vert_y$$ be if $$f(x,y,z) = x^2 + 5y + z^3$$

is it $$\frac{\partial f}{\partial x}\vert_y = 2x + 3z^2\frac{d z}{dx}$$

or something else?

• Your guess is sort of correct, if $z = z(x)$ then what you wrote is correct but if $z = z(x, ...)$ then you need to write $\frac{\partial z}{\partial x}$ instead of $\frac{d z}{d x}$ and it needs to be evident what variables are being held constant in the partial derivative. This can be done by writing $\left( \frac{\partial z}{\partial x} \right)_{ab}$ where or just by writing out $z$ as an explicit function like $z(x, a, b) = x^2 + x a + b$ as the case may be. – jcarpenter2 May 2 '20 at 14:21

All variables other than $$x$$ in your example must be held constant. The ambiguity arises from definitions like
$$f = x + y = 2 x + z$$
Here it is evident that $$y = x + z$$ but it is not specified whether $$f$$ is a function of $$(x, y)$$ or $$(x, z)$$ if you were to write $$\frac{\partial f}{\partial x}$$. Since there is nothing to say which is the more fundamental quantity, $$y$$ or $$z$$ (or if it even makes sense to ask that question), "there is nowhere to stand" (as the Buddhists say), and you must specify which variable, $$y$$ or $$z$$, you are choosing to hold constant.
$$\left( \frac{\partial f}{\partial x} \right)_y = 1$$
$$\left( \frac{\partial f}{\partial x} \right)_z = 2$$