Deriving marginal effects in multinomial logit model For the multinomial logit model, it holds that:
$$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$.
Now my book states that the marginal effect is as follows:
$$\dfrac{\partial \operatorname{P}[y_i = j]}{\partial x_{ij}} = \operatorname{P}[y_i=j](1-\operatorname{P}[y_i=j])\beta_1$$
I tried derving this but I did not find an easy way. Could anyone please help? Thanks in advance.

Cross multiply the equation to obtain:
$$\operatorname{P}(y_i=j)\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})=\exp(\beta_{0,j} + \beta_1 x_{ij})$$
Then deriving with respect to $x_{ij}$ on both sides of the equality gives the following:
$$\dfrac{\partial \operatorname{P}(y_i = j)}{\partial x_{ij}}\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})+\operatorname{P}(y_i=j)\beta_1  \exp(\beta_{0,j} + \beta_1 x_{ij})= \beta_1\exp(\beta_{0,j} + \beta_1 x_{ij})$$
Now if we divide by $\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})$ we obtain:
$$\dfrac{\partial \operatorname{P}(y_i = j)}{\partial x_{ij}}+\operatorname{P}(y_i=j)\beta_1  \operatorname{P}(y_i=j)= \beta_1\operatorname{P}(y_i=j)$$
And the result is complete.
