derivative of x^3(x-5^2)

解

\frac{d}{d x} (x^{3} (x - 5)^{2})

5 x^{4} - 40 x^{3} + 75 x^{2}

解答ステップ

\frac{d}{d x} (x^{3} (x - 5)^{2})

積の法則を適用:

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

f = x^{3}, g = (x - 5)^{2}

= \frac{d}{d x} (x^{3}) (x - 5)^{2} + \frac{d}{d x} ((x - 5)^{2}) x^{3}

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

\frac{d}{d x} ((x - 5)^{2}) = 2 (x - 5)

= 3 x^{2} (x - 5)^{2} + 2 (x - 5) x^{3}

簡素化

3 x^{2} (x - 5)^{2} + 2 (x - 5) x^{3} : 5 x^{4} - 40 x^{3} + 75 x^{2}

= 5 x^{4} - 40 x^{3} + 75 x^{2}