(\partial)/(\partial x)(1/(f(x)^'(f(x)^{-1)(x))})

解

\frac{\partial}{\partial x} \frac{1}{f ( x ) x ^{^{'}} ( f ( x ) ( x ) ^{- 1} ( x ) )}

- \frac{' f ( x ) + 2 x \frac{\partial}{\partial x} ( f ( x ) )}{x ^{1 +'} f ( x ) ^{3}}

解答ステップ

\frac{\partial}{\partial x} \frac{1}{f ( x ) x ^{^{'}} ( f ( x ) ( x ) ^{- 1} ( x ) )}

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= \frac{\partial}{\partial x} ((f (x) x^{^{^{'}}} f (x) x^{- 1} x)^{- 1})

連鎖法則を適用:

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

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

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

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

簡素化

- \frac{1}{( f ( x ) x ^{^{'}} f ( x ) x ^{- 1} x ) ^{2}} (^{'} x^{^{'} - 1} f (x)^{2} + 2 f (x) \frac{\partial}{\partial x} (f (x)) x^{^{'}}) : - \frac{^{'} f ( x ) + 2 x \frac{\partial}{\partial x} ( f ( x ) )}{x ^{1 +^{'}} f ( x ) ^{3}}

= - \frac{^{'} f ( x ) + 2 x \frac{\partial}{\partial x} ( f ( x ) )}{x ^{1 +^{'}} f ( x ) ^{3}}