derivative of (sec(sin(sqrt(x^2+6^x)))^4)

解

\frac{d}{d x} ((sec (sin (x^{2} + 6^{x})))^{4})

\frac{2 sec ^{4} ( sin ( x ^{2} + 6 ^{x} ) ) cos ( x ^{2} + 6 ^{x} ) tan ( sin ( x ^{2} + 6 ^{x} ) ) ( 2 x + ln ( 6 ) \cdot 6 ^{x} )}{x ^{2} + 6 ^{x}}

解答ステップ

\frac{d}{d x} ((sec (sin (x^{2} + 6^{x})))^{4})

連鎖法則を適用:

4 (sec (sin (x^{2} + 6^{x})))^{3} \frac{d}{d x} (sec (sin (x^{2} + 6^{x})))

= 4 (sec (sin (x^{2} + 6^{x})))^{3} \frac{d}{d x} (sec (sin (x^{2} + 6^{x})))

\frac{d}{d x} (sec (sin (x^{2} + 6^{x}))) = \frac{cos ( x ^{2} + 6 ^{x} ) sec ( sin ( x ^{2} + 6 ^{x} ) ) tan ( sin ( x ^{2} + 6 ^{x} ) ) ( 2 x + ln ( 6 ) \cdot 6 ^{x} )}{2 x ^{2} + 6 ^{x}}

= 4 (sec (sin (x^{2} + 6^{x})))^{3} \frac{cos ( x ^{2} + 6 ^{x} ) sec ( sin ( x ^{2} + 6 ^{x} ) ) tan ( sin ( x ^{2} + 6 ^{x} ) ) ( 2 x + ln ( 6 ) \cdot 6 ^{x} )}{2 x ^{2} + 6 ^{x}}

簡素化

4 sec^{3} (sin (x^{2} + 6^{x})) \frac{cos ( x ^{2} + 6 ^{x} ) sec ( sin ( x ^{2} + 6 ^{x} ) ) tan ( sin ( x ^{2} + 6 ^{x} ) ) ( 2 x + ln ( 6 ) \cdot 6 ^{x} )}{2 x ^{2} + 6 ^{x}} : \frac{2 sec ^{4} ( sin ( x ^{2} + 6 ^{x} ) ) cos ( x ^{2} + 6 ^{x} ) tan ( sin ( x ^{2} + 6 ^{x} ) ) ( 2 x + ln ( 6 ) \cdot 6 ^{x} )}{x ^{2} + 6 ^{x}}

= \frac{2 sec ^{4} ( sin ( x ^{2} + 6 ^{x} ) ) cos ( x ^{2} + 6 ^{x} ) tan ( sin ( x ^{2} + 6 ^{x} ) ) ( 2 x + ln ( 6 ) \cdot 6 ^{x} )}{x ^{2} + 6 ^{x}}