(\partial)/(\partial x)((tx(1-x))/(1-t^2))

解

\frac{\partial}{\partial x} (\frac{t x ( 1 - x )}{1 - t ^{2}})

\frac{t ( 1 - 2 x )}{1 - t ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{t x ( 1 - x )}{1 - t ^{2}})

t

を定数として扱う

定数を除去:

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

= \frac{t}{1 - t ^{2}} \frac{\partial}{\partial x} (x (1 - x))

積の法則を適用:

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

f = x, g = 1 - x

= \frac{t}{1 - t ^{2}} (\frac{\partial}{\partial x} (x) (1 - x) + \frac{\partial}{\partial x} (1 - x) x)

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

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

= \frac{t}{1 - t ^{2}} (1 \cdot (1 - x) + (- 1) x)

簡素化

\frac{t}{1 - t ^{2}} (1 \cdot (1 - x) + (- 1) x) : \frac{t ( 1 - 2 x )}{1 - t ^{2}}

= \frac{t ( 1 - 2 x )}{1 - t ^{2}}