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素微分
,\end{align*}$$
などが分かる。
===積・商の素微分===
$$\begin{align*}
& \left[\left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{q_k}\right)\right]' = \left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{q_k}\right)\left(\sum_{j=1}^n \frac{q_j}{p_j} + \sum_{j={n+1}}^{\infty} \frac{q_j}{p_j} \right) \\
=& \left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{q_k}\right)\left(\sum_{j=1}^n \frac{q_j}{p_j} \right) + \left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{q_k}\right)\left(\sum_{j={n+1}}^{\infty} \frac{q_j}{p_j} \right) = \left(\prod_{k=1}^n {p_k}^{q_k}\right)'\left(\prod_{k={n+1}}^{\infty} {p_k}^{q_k}\right) + \left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{q_k}\right)' \\
=& \frac{\left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{-q_k}\right)\left(\sum_{j=1}^n \frac{q_j}{p_j} \right) - \left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{-q_k}\right)\left(\sum_{j={n+1}}^{\infty} \frac{-q_j}{p_j} \right)}{\left(\prod_{k={n+1}}^{\infty} {p_k}^{-q_k}\right)^2} = \frac{\left(\prod_{k=1}^n {p_k}^{q_k}\right)'\left(\prod_{k={n+1}}^{\infty} {p_k}^{-q_k}\right) - \left(\prod_{k=1}^n {p_k}^{q_k}\right)\left(\prod_{k={n+1}}^{\infty} {p_k}^{-q_k}\right)'}{\left(\prod_{k={n+1}}^{\infty} {p_k}^{-q_k}\right)^2}
\end{align*}$$
== 拡張 ==