Unknown identity

Question

$$e^f(\frac{\partial}{\partial x})g=(\frac{\partial}{\partial x}-\frac{\partial f}{\partial x})g$$

What identity is this, cause it is not stated from here(page 6 above eqn 24)?

Answer 1

As stated it is not an identity. There is a typo in the notes. The identity used in fact is

$$ e^f \partial_x g = (\partial_x - \partial_x f) (e^f g) $$

which is a direct consequence of the product rule of elementary calculus.

Answer 2

There is a typo I think. If you write

$\frac{\partial (e^{f}g)}{\partial x}=ge^{f}\frac{\partial f}{\partial x}+e^{f}\frac{\partial g}{\partial x}$ and rearrange, you get

$e^{f}\frac{\partial g}{\partial x}=\frac{\partial (e^{f}g)}{\partial x}-ge^{f}\frac{\partial f}{\partial x}$,

which is not equal to the stated item.