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) ?

Thank you for your time.