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

해법

\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))))

솔루션 단계

\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))))