derivative of {f}(x{f}(x)(x^2))

解

\frac{d}{d x} (f (x) f (x) (x^{2}))

2 x f (x)^{2} + 2 f (x) \frac{d}{d x} (f (x)) x^{2}

解答ステップ

\frac{d}{d x} (f (x) f (x) (x^{2}))

簡素化

f (x) f (x) (x^{2}) : x^{2} f (x)^{2}

= \frac{d}{d x} (x^{2} f (x)^{2})

積の法則を適用:

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

f = x^{2}, g = f (x)^{2}

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

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

\frac{d}{d x} (f (x)^{2}) = 2 f (x) \frac{d}{d x} (f (x))

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