(d^2)/(dx^2)((y)(x^2+y^2-6x-8y+16))

Lösung

\frac{d ^{2}}{d x ^{2}} ((y) (x^{2} + y^{2} - 6 x - 8 y + 16))

x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} - 6 x \frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 2 y + 6 y \frac{d y}{d x}^{2} + 3 y^{2} \frac{d ^{2} y}{d x ^{2}} - 16 \frac{d y}{d x}^{2} - 16 y \frac{d ^{2} y}{d x ^{2}} + 16 \frac{d ^{2} y}{d x ^{2}}

Schritte zur Lösung

\frac{d ^{2}}{d x ^{2}} ((y) (x^{2} + y^{2} - 6 x - 8 y + 16))

\frac{d}{d x} (y (x^{2} + y^{2} - 6 x - 8 y + 16)) = x^{2} \frac{d y}{d x} - 6 x \frac{d y}{d x} + 2 x y + 3 y^{2} \frac{d y}{d x} - 16 y \frac{d y}{d x} + 16 \frac{d y}{d x} - 6 y

= \frac{d}{d x} (x^{2} \frac{d y}{d x} - 6 x \frac{d y}{d x} + 2 x y + 3 y^{2} \frac{d y}{d x} - 16 y \frac{d y}{d x} + 16 \frac{d y}{d x} - 6 y)

\frac{d}{d x} (x^{2} \frac{d y}{d x} - 6 x \frac{d y}{d x} + 2 x y + 3 y^{2} \frac{d y}{d x} - 16 y \frac{d y}{d x} + 16 \frac{d y}{d x} - 6 y) = x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} - 6 x \frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 2 y + 6 y \frac{d y}{d x}^{2} + 3 y^{2} \frac{d ^{2} y}{d x ^{2}} - 16 \frac{d y}{d x}^{2} - 16 y \frac{d ^{2} y}{d x ^{2}} + 16 \frac{d ^{2} y}{d x ^{2}}

= x^{2} \frac{d ^{2} y}{d x ^{2}} + 4 x \frac{d y}{d x} - 6 x \frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 2 y + 6 y \frac{d y}{d x}^{2} + 3 y^{2} \frac{d ^{2} y}{d x ^{2}} - 16 \frac{d y}{d x}^{2} - 16 y \frac{d ^{2} y}{d x ^{2}} + 16 \frac{d ^{2} y}{d x ^{2}}

e x p an d (\frac{1}{x ^{2}} - 3 x)^{5}

e x p an d \frac{x - 5}{( x - 1 ) ( x + 4 )}

s im pl i f y (2 x + (4 x^{2} - 1))^{2}

e x p an d (2 x^{2} + 6) (3 x^{\frac{1}{2}})

e x p an d (x^{2 + y^{2}})^{2}