derivative of (x^2+3^{sin(x)})

解

\frac{d}{d x} ((x^{2} + 3)^{s i n (x)})

(cos (x) ln (x^{2} + 3) + \frac{2 x sin ( x )}{x ^{2} + 3}) (x^{2} + 3)^{s i n (x)}

解答ステップ

\frac{d}{d x} ((x^{2} + 3)^{s i n (x)})

指数の規則を適用する:

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

(x^{2} + 3)^{s i n (x)} = e^{s i n (x) l n (x^{2} + 3)}

= \frac{d}{d x} (e^{s i n (x) l n (x^{2} + 3)})

連鎖法則を適用:

e^{s i n (x) l n (x^{2} + 3)} \frac{d}{d x} (sin (x) ln (x^{2} + 3))

= e^{s i n (x) l n (x^{2} + 3)} \frac{d}{d x} (sin (x) ln (x^{2} + 3))

\frac{d}{d x} (sin (x) ln (x^{2} + 3)) = cos (x) ln (x^{2} + 3) + \frac{2 x sin ( x )}{x ^{2} + 3}

= e^{s i n (x) l n (x^{2} + 3)} (cos (x) ln (x^{2} + 3) + \frac{2 x sin ( x )}{x ^{2} + 3})

e^{s i n (x) l n (x^{2} + 3)} = (x^{2} + 3)^{s i n (x)}

= (cos (x) ln (x^{2} + 3) + \frac{2 x sin ( x )}{x ^{2} + 3}) (x^{2} + 3)^{s i n (x)}