derivative of ln(3^xsec^3(4x))

解

\frac{d}{d x} (ln (3^{x} sec^{3} (4 x)))

ln (3) + 12 tan (4 x)

解答ステップ

\frac{d}{d x} (ln (3^{x} sec^{3} (4 x)))

連鎖法則を適用:

\frac{1}{3 ^{x} sec ^{3} ( 4 x )} \frac{d}{d x} (3^{x} sec^{3} (4 x))

= \frac{1}{3 ^{x} sec ^{3} ( 4 x )} \frac{d}{d x} (3^{x} sec^{3} (4 x))

\frac{d}{d x} (3^{x} sec^{3} (4 x)) = ln (3) \cdot 3^{x} sec^{3} (4 x) + 4 \cdot 3^{1 + x} sec^{3} (4 x) tan (4 x)

= \frac{1}{3 ^{x} sec ^{3} ( 4 x )} (ln (3) \cdot 3^{x} sec^{3} (4 x) + 4 \cdot 3^{1 + x} sec^{3} (4 x) tan (4 x))

簡素化

\frac{1}{3 ^{x} sec ^{3} ( 4 x )} (ln (3) \cdot 3^{x} sec^{3} (4 x) + 4 \cdot 3^{1 + x} sec^{3} (4 x) tan (4 x)) : ln (3) + 12 tan (4 x)

= ln (3) + 12 tan (4 x)