derivative of (x^3y+xy^3/(x^2+y^2))

解答

\frac{d}{d x} (\frac{x ^{3} y + x y ^{3}}{x ^{2} + y ^{2}})

y + x \frac{d y}{d x}

求解步骤

\frac{d}{d x} (\frac{x ^{3} y + x y ^{3}}{x ^{2} + y ^{2}})

化简

\frac{x ^{3} y + x y ^{3}}{x ^{2} + y ^{2}} : x y

= \frac{d}{d x} (x y)

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = y

= \frac{d x}{d x} y + \frac{d y}{d x} x

\frac{d x}{d x} = 1

= 1 \cdot y + \frac{d y}{d x} x

化简

= y + x \frac{d y}{d x}

t an g e n t f (x) = x - 4, (13, 3)

t an g e n t f (x) = \frac{5}{x + 1}, at x = \frac{- 1}{3}

x \to 0 lim (\frac{0 + 5}{3 ( 0 )})

d er i v a t i v e f (x) = \frac{x ^{2} + 8 x + 3}{x}

x \to \infty lim (\frac{x ^{2} + 2 x + 5}{e ^{3 x}})