Say you have a log-level regression as follows:(adsbygoogle = window.adsbygoogle || []).push({});

$$\log Y = \beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n$$

We're trying come up with a meaningful interpretation for changes Y due to a change in some X_{k}.

If we take the partial derivative with respect to X_{k}. we end up with

$$\frac{dY}{Y} = \beta_k \cdot dX_k$$

which implies that if X_{k}. increases by 1, you expect Y to increase by 100β_{k}percent.

Can someone walk through the calculus to get from this

$$\frac{\partial}{\partial X_k} \log{y}= \frac{\partial}{\partial X_k} (\beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n)$$

to this

$$\frac{dY}{Y} = \beta_k dX_k$$?

I'm particularly confused about how one transitions from a partial derivate to a total derivative.

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# A Differential of Multiple Linear Regression

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