derivative of (x^2+y^2ln(x+y))

解

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

ln (x + y) (2 x + 2 y \frac{d y}{d x}) + \frac{( 1 + \frac{d y}{d x} ) ( x ^{2} + y ^{2} )}{x + y}

解答ステップ

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

積の法則を適用:

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

f = x^{2} + y^{2}, g = ln (x + y)

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

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

\frac{d}{d x} (ln (x + y)) = \frac{1 + \frac{d y}{d x}}{x + y}

= (2 x + 2 y \frac{d y}{d x}) ln (x + y) + \frac{1 + \frac{d y}{d x}}{x + y} (x^{2} + y^{2})

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= ln (x + y) (2 x + 2 y \frac{d y}{d x}) + \frac{( 1 + \frac{d y}{d x} ) ( x ^{2} + y ^{2} )}{x + y}