derivative f(x)=(5^x)/(cos(x))

解

導関数

f (x) = \frac{5 ^{x}}{cos ( x )}

5^{x} ln (5) sec (x) + sec (x) tan (x) \cdot 5^{x}

解答ステップ

\frac{d}{d x} (\frac{5 ^{x}}{cos ( x )})

簡素化

\frac{5 ^{x}}{cos ( x )} : 5^{x} sec (x)

= \frac{d}{d x} (5^{x} sec (x))

積の法則を適用:

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

f = 5^{x}, g = sec (x)

= \frac{d}{d x} (5^{x}) sec (x) + \frac{d}{d x} (sec (x)) 5^{x}

\frac{d}{d x} (5^{x}) = 5^{x} ln (5)

\frac{d}{d x} (sec (x)) = sec (x) tan (x)

= 5^{x} ln (5) sec (x) + sec (x) tan (x) \cdot 5^{x}