(\partial)/(\partial y)(ln(1+x^2+e^y+z))

解

\frac{\partial}{\partial y} (ln (1 + x^{2} + e^{y} + z))

\frac{e ^{y}}{1 + x ^{2} + e ^{y} + z}

解答ステップ

\frac{\partial}{\partial y} (ln (1 + x^{2} + e^{y} + z))

x, z

を定数として扱う

連鎖法則を適用:

\frac{1}{1 + x ^{2} + e ^{y} + z} \frac{\partial}{\partial y} (1 + x^{2} + e^{y} + z)

= \frac{1}{1 + x ^{2} + e ^{y} + z} \frac{\partial}{\partial y} (1 + x^{2} + e^{y} + z)

\frac{\partial}{\partial y} (1 + x^{2} + e^{y} + z) = e^{y}

= \frac{1}{1 + x ^{2} + e ^{y} + z} e^{y}

簡素化

\frac{1}{1 + x ^{2} + e ^{y} + z} e^{y} : \frac{e ^{y}}{1 + x ^{2} + e ^{y} + z}

= \frac{e ^{y}}{1 + x ^{2} + e ^{y} + z}