(\partial)/(\partial x)(A^3(x/y)^2(1-x))

解

\frac{\partial}{\partial x} (A^{3} (\frac{x}{y})^{2} (1 - x))

\frac{A ^{3} ( 2 x - 3 x ^{2} )}{y ^{2}}

解答ステップ

\frac{\partial}{\partial x} (A^{3} (\frac{x}{y})^{2} (1 - x))

A, y

を定数として扱う

定数を除去:

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

= A^{3} \frac{\partial}{\partial x} ((\frac{x}{y})^{2} (1 - x))

積の法則を適用:

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

f = (\frac{x}{y})^{2}, g = 1 - x

= A^{3} (\frac{\partial}{\partial x} ((\frac{x}{y})^{2}) (1 - x) + \frac{\partial}{\partial x} (1 - x) (\frac{x}{y})^{2})

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

\frac{\partial}{\partial x} (1 - x) = - 1

= A^{3} (\frac{2 x}{y ^{2}} (1 - x) + (- 1) (\frac{x}{y})^{2})

簡素化

A^{3} (\frac{2 x}{y ^{2}} (1 - x) + (- 1) (\frac{x}{y})^{2}) : \frac{A ^{3} ( 2 x - 3 x ^{2} )}{y ^{2}}

= \frac{A ^{3} ( 2 x - 3 x ^{2} )}{y ^{2}}