derivative (t+1)^3sqrt(t^2+84)

解

導関数

(t + 1)^{3} t^{2} + 84

\frac{3 t ^{4} + 6 t ^{3} + 255 t ^{2} + t ( t + 1 ) ^{3} + 504 t + 252}{t ^{2} + 84}

解答ステップ

\frac{d}{d t} ((t + 1)^{3} t^{2} + 84)

積の法則を適用:

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

f = (t + 1)^{3}, g = t^{2} + 84

= \frac{d}{d t} ((t + 1)^{3}) t^{2} + 84 + \frac{d}{d t} (t^{2} + 84) (t + 1)^{3}

\frac{d}{d t} ((t + 1)^{3}) = 3 (t + 1)^{2}

\frac{d}{d t} (t^{2} + 84) = \frac{t}{t ^{2} + 84}

= 3 (t + 1)^{2} t^{2} + 84 + \frac{t}{t ^{2} + 84} (t + 1)^{3}

簡素化

3 (t + 1)^{2} t^{2} + 84 + \frac{t}{t ^{2} + 84} (t + 1)^{3} : \frac{3 t ^{4} + 6 t ^{3} + 255 t ^{2} + t ( t + 1 ) ^{3} + 504 t + 252}{t ^{2} + 84}

= \frac{3 t ^{4} + 6 t ^{3} + 255 t ^{2} + t ( t + 1 ) ^{3} + 504 t + 252}{t ^{2} + 84}