d/(d{x)}(arcsin^4({x}^7{y}^2+({z})/({x)^3}))

解

\frac{d}{d x} (arcsin^{4} (x^{7} y^{2} + \frac{z}{x ^{3}}))

\frac{4 arcsin ^{3} ( \frac{y ^{2} x ^{10} + z}{x ^{3}} ) ( 7 y ^{2} x ^{10} - 3 z )}{x x ^{6} - x ^{20} y ^{4} - 2 x ^{10} y ^{2} z - z ^{2}}

解答ステップ

\frac{d}{d x} (arcsin^{4} (x^{7} y^{2} + \frac{z}{x ^{3}}))

連鎖法則を適用:

4 (arcsin (x^{7} y^{2} + \frac{z}{x ^{3}}))^{3} \frac{d}{d x} (arcsin (x^{7} y^{2} + \frac{z}{x ^{3}}))

= 4 (arcsin (x^{7} y^{2} + \frac{z}{x ^{3}}))^{3} \frac{d}{d x} (arcsin (x^{7} y^{2} + \frac{z}{x ^{3}}))

\frac{d}{d x} (arcsin (x^{7} y^{2} + \frac{z}{x ^{3}})) = \frac{x ^{6}}{x ^{6} - x ^{20} y ^{4} - 2 x ^{10} y ^{2} z - z ^{2}} (7 x^{6} y^{2} - \frac{3 z}{x ^{4}})

= 4 (arcsin (x^{7} y^{2} + \frac{z}{x ^{3}}))^{3} \frac{x ^{6}}{x ^{6} - x ^{20} y ^{4} - 2 x ^{10} y ^{2} z - z ^{2}} (7 x^{6} y^{2} - \frac{3 z}{x ^{4}})

簡素化

4 arcsin^{3} (x^{7} y^{2} + \frac{z}{x ^{3}}) \frac{x ^{6}}{x ^{6} - x ^{20} y ^{4} - 2 x ^{10} y ^{2} z - z ^{2}} (7 x^{6} y^{2} - \frac{3 z}{x ^{4}}) : \frac{4 arcsin ^{3} ( \frac{y ^{2} x ^{10} + z}{x ^{3}} ) ( 7 y ^{2} x ^{10} - 3 z )}{x x ^{6} - x ^{20} y ^{4} - 2 x ^{10} y ^{2} z - z ^{2}}

= \frac{4 arcsin ^{3} ( \frac{y ^{2} x ^{10} + z}{x ^{3}} ) ( 7 y ^{2} x ^{10} - 3 z )}{x x ^{6} - x ^{20} y ^{4} - 2 x ^{10} y ^{2} z - z ^{2}}