## Partial Derivative Property

I was rereading my Calculus book when I saw this property

For $\alpha \in \mathbb{R} \neq 0$ and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ a continuously differentiable function.

$\frac{\partial f}{ \partial (\alpha p)} (x) = \alpha \frac{\partial f }{\partial p} (x)$
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I can see that this property holds true for $\alpha = 1$.

If $\alpha \neq 1$, then would

$\frac{\partial f }{\partial (\alpha p)}(x) = \sum_{i=1}^{n} \alpha p_{i} \frac{\partial f}{ \partial x_{i}} (x) = \alpha \frac{\partial f }{\partial p} (x)$?