How to prove the monotonicity of Softmax when decreasing the weighted inputs? Softmax function $a^L_j = z^L_j / sum_k(z^L_k)$. When we think about the monotonicity of the Softmax function, $∂a^L_j/∂z^L_k$ is positive if j=k, and negative if j≠k. As a consequence, increasing $z^L_j$ is guaranteed to increase the corresponding output activation $a^L_j$, and will decrease all the other output activations.
My question is, decreasing $z^L_j$ will decrease the corresponding output and will increase all the other output activations. But, $\frac{∂a^L_j}{∂z^L_k}$ is always positive if j=k, and negative if j≠k. So, I'm curious how to prove that decreasing $z^L_j$ will actually decrease the corresponding output when we always have $\frac{∂a^L_j}{∂z^L_k}$ is positive when j=k?

(Consolidating my comments into an answer.)
This comes directly from the definition of a partial derivative.
A partial derivative is the (instantaneous) rate of change of one variable with respect to another. With a positive partial derivative, you increase $z$ by some amount, you increase $a$ by a corresponding amount. Similarly, if you decrease $z$, then you decrease $a$ by a corresponding amount.
