Manipulating Partial Derivatives

2.3. Manipulating Partial Derivatives#

We sometimes need to find derivatives we don’t obtain easily from one of these potentials. For this, we use some calculus rules.

If x(y, z) and y(x, z), then:

\[\left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z}\]

Example:

\[Let: x = \frac{y^2}{z}\]

We can rearrange this to:

\[y = \pm \sqrt{xz}\]

Now compute:

\[\left( \frac{\partial x}{\partial y} \right)_z = \frac{2y}{z}\]

Then:

\[\left( \frac{\partial y}{\partial x} \right)_z = \frac{z}{2y}\]

And confirm the inversion:

\[\frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} = \frac{2y}{z}\]

\[\left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z\]

Example:

\[ w = yz \]

So,

\[y = \frac{w}{z}\]

Then:

\[x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3}\]

Now compute:

\[\left( \frac{\partial x}{\partial w} \right)_z = \frac{2w}{z^3}\]

And:

\[\left( \frac{\partial w}{\partial y} \right)_z = z\]

Therefore:

\[\left( \frac{\partial x}{\partial y} \right)_z = \frac{2w}{z^3} \cdot z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z}\]