(d^2)/(dx^2)((z)(x^2+y^2+z^2-c^2))

解

\frac{d ^{2}}{d x ^{2}} ((z (x)) (x^{2} + y^{2} + z (x)^{2} - c^{2}))

x^{2} \frac{d ^{2} z}{d x ^{2}} + 4 x \frac{d z}{d x} - c^{2} \frac{d ^{2} z}{d x ^{2}} + 2 z + 4 y \frac{d z}{d x} \frac{d y}{d x} + y^{2} \frac{d ^{2} z}{d x ^{2}} + 6 z \frac{d z}{d x}^{2} + 3 z^{2} \frac{d ^{2} z}{d x ^{2}} + 2 z \frac{d y}{d x}^{2} + 2 zy \frac{d ^{2} y}{d x ^{2}}

解答ステップ

\frac{d ^{2}}{d x ^{2}} ((z) (x^{2} + y^{2} + z^{2} - c^{2}))

\frac{d}{d x} (z (x^{2} + y^{2} + z^{2} - c^{2})) = x^{2} \frac{d z}{d x} + 2 x z - c^{2} \frac{d z}{d x} + y^{2} \frac{d z}{d x} + 3 z^{2} \frac{d z}{d x} + 2 zy \frac{d y}{d x}

= \frac{d}{d x} (x^{2} \frac{d z}{d x} + 2 x z - c^{2} \frac{d z}{d x} + y^{2} \frac{d z}{d x} + 3 z^{2} \frac{d z}{d x} + 2 zy \frac{d y}{d x})

\frac{d}{d x} (x^{2} \frac{d z}{d x} + 2 x z - c^{2} \frac{d z}{d x} + y^{2} \frac{d z}{d x} + 3 z^{2} \frac{d z}{d x} + 2 zy \frac{d y}{d x}) = x^{2} \frac{d ^{2} z}{d x ^{2}} + 4 x \frac{d z}{d x} - c^{2} \frac{d ^{2} z}{d x ^{2}} + 2 z + 4 y \frac{d z}{d x} \frac{d y}{d x} + y^{2} \frac{d ^{2} z}{d x ^{2}} + 6 z \frac{d z}{d x}^{2} + 3 z^{2} \frac{d ^{2} z}{d x ^{2}} + 2 z \frac{d y}{d x}^{2} + 2 zy \frac{d ^{2} y}{d x ^{2}}

= x^{2} \frac{d ^{2} z}{d x ^{2}} + 4 x \frac{d z}{d x} - c^{2} \frac{d ^{2} z}{d x ^{2}} + 2 z + 4 y \frac{d z}{d x} \frac{d y}{d x} + y^{2} \frac{d ^{2} z}{d x ^{2}} + 6 z \frac{d z}{d x}^{2} + 3 z^{2} \frac{d ^{2} z}{d x ^{2}} + 2 z \frac{d y}{d x}^{2} + 2 zy \frac{d ^{2} y}{d x ^{2}}