(\partial)/(\partial x)(5(y^2-x^2)ln(x+y))

解

\frac{\partial}{\partial x} (5 (y^{2} - x^{2}) ln (x + y))

5 (- 2 x ln (x + y) + y - x)

解答ステップ

\frac{\partial}{\partial x} (5 (y^{2} - x^{2}) ln (x + y))

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{\partial}{\partial x} ((y^{2} - x^{2}) ln (x + y))

積の法則を適用:

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

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

= 5 (\frac{\partial}{\partial x} (y^{2} - x^{2}) ln (x + y) + \frac{\partial}{\partial x} (ln (x + y)) (y^{2} - x^{2}))

\frac{\partial}{\partial x} (y^{2} - x^{2}) = - 2 x

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

= 5 ((- 2 x) ln (x + y) + \frac{1}{x + y} (y^{2} - x^{2}))

簡素化

5 ((- 2 x) ln (x + y) + \frac{1}{x + y} (y^{2} - x^{2})) : 5 (- 2 x ln (x + y) + y - x)

= 5 (- 2 x ln (x + y) + y - x)