derivative of (sin(x^2+1)^{1/({f)(x)}})

解

\frac{d}{d x} ((sin (x^{2}) + 1)^{\frac{1}{f ( x )}})

e^{\frac{l n ( s i n ( x ^{2} ) + 1 )}{f ( x )}} (- \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) f ( x )})

解答ステップ

\frac{d}{d x} ((sin (x^{2}) + 1)^{\frac{1}{f ( x )}})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

(sin (x^{2}) + 1)^{\frac{1}{f ( x )}} = e^{\frac{1}{f ( x )} l n (s i n (x^{2}) + 1)}

= \frac{d}{d x} (e^{\frac{1}{f ( x )} l n (s i n (x^{2}) + 1)})

連鎖法則を適用:

e^{\frac{1}{f ( x )} l n (s i n (x^{2}) + 1)} \frac{d}{d x} (\frac{1}{f ( x )} ln (sin (x^{2}) + 1))

= e^{\frac{1}{f ( x )} l n (s i n (x^{2}) + 1)} \frac{d}{d x} (\frac{1}{f ( x )} ln (sin (x^{2}) + 1))

\frac{d}{d x} (\frac{1}{f ( x )} ln (sin (x^{2}) + 1)) = - \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) f ( x )}

= e^{\frac{1}{f ( x )} l n (s i n (x^{2}) + 1)} (- \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) f ( x )})

乗じる

\frac{1}{f ( x )} ln (sin (x^{2}) + 1) : \frac{ln ( sin ( x ^{2} ) + 1 )}{f ( x )}

= e^{\frac{l n ( s i n ( x ^{2} ) + 1 )}{f ( x )}} (- \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) f ( x )})