(d^2)/(dx^2)(({g}(x,y))(x^2y+cos(y)+y,x))

Lösung

\frac{d ^{2}}{d x ^{2}} ((g (x, y)) (x^{2} y + cos (y) + y, x))

(\frac{d ^{2}}{d x ^{2}} (g (x, y)) (x^{2} y + cos (y) + y, x) + (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))) \frac{d}{d x} (g (x, y))) + ((2 (y + x \frac{d y}{d x}) + \frac{d ^{2} y}{d x ^{2}} x^{2} + 2 x \frac{d y}{d x} - (cos (y) \frac{d y}{d x}^{2} + sin (y) \frac{d ^{2} y}{d x ^{2}}) + (, (x \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} + \frac{d y}{d x}))) g (x, y) + \frac{d}{d x} (g (x, y)) (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))))

Schritte zur Lösung

\frac{d ^{2}}{d x ^{2}} ((g (x, y)) (x^{2} y + cos (y) + y, x))

\frac{d}{d x} (g (x, y) (x^{2} y + cos (y) + y, x)) = \frac{d}{d x} (g (x, y)) (x^{2} y + cos (y) + y, x) + (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))) g (x, y)

= \frac{d}{d x} (\frac{d}{d x} (g (x, y)) (x^{2} y + cos (y) + y, x) + (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))) g (x, y))

\frac{d}{d x} (\frac{d}{d x} (g (x, y)) (x^{2} y + cos (y) + y, x) + (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))) g (x, y)) = (\frac{d ^{2}}{d x ^{2}} (g (x, y)) (x^{2} y + cos (y) + y, x) + (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))) \frac{d}{d x} (g (x, y))) + ((2 (y + x \frac{d y}{d x}) + \frac{d ^{2} y}{d x ^{2}} x^{2} + 2 x \frac{d y}{d x} - (cos (y) \frac{d y}{d x}^{2} + sin (y) \frac{d ^{2} y}{d x ^{2}}) + (, (x \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} + \frac{d y}{d x}))) g (x, y) + \frac{d}{d x} (g (x, y)) (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))))

= (\frac{d ^{2}}{d x ^{2}} (g (x, y)) (x^{2} y + cos (y) + y, x) + (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))) \frac{d}{d x} (g (x, y))) + ((2 (y + x \frac{d y}{d x}) + \frac{d ^{2} y}{d x ^{2}} x^{2} + 2 x \frac{d y}{d x} - (cos (y) \frac{d y}{d x}^{2} + sin (y) \frac{d ^{2} y}{d x ^{2}}) + (, (x \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} + \frac{d y}{d x}))) g (x, y) + \frac{d}{d x} (g (x, y)) (2 x y + \frac{d y}{d x} x^{2} - sin (y) \frac{d y}{d x} + (, (\frac{d y}{d x} x + 1 \cdot y))))

e x p an d (5 4 x + x) (2 x^{3} - 3 x^{2} + x - 1)

e x p an d (5 x^{3} + x^{2} + x) (1.5 x^{2} + 3)

s im pl i f y (3 + x + 2)^{2}

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

e x p an d (- 2)^{6}