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

解

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

2 x f (x)^{3} + 3 f (x)^{2} \frac{d}{d x} (f (x)) x^{2}

解答ステップ

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

簡素化

f (x) (x f (x) (x f (x))) : x^{2} f (x)^{3}

= \frac{d}{d x} (x^{2} f (x)^{3})

積の法則を適用:

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

f = x^{2}, g = f (x)^{3}

= \frac{d}{d x} (x^{2}) f (x)^{3} + \frac{d}{d x} (f (x)^{3}) x^{2}

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

\frac{d}{d x} (f (x)^{3}) = 3 f (x)^{2} \frac{d}{d x} (f (x))

= 2 x f (x)^{3} + 3 f (x)^{2} \frac{d}{d x} (f (x)) x^{2}