derivative of x^{5cos(x})

解

\frac{d}{d x} (x^{5 c o s (x)})

5 x^{5 c o s (x)} (- sin (x) ln (x) + \frac{cos ( x )}{x})

解答ステップ

\frac{d}{d x} (x^{5 c o s (x)})

指数の規則を適用する:

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

x^{5 c o s (x)} = e^{5 c o s (x) l n (x)}

= \frac{d}{d x} (e^{5 c o s (x) l n (x)})

連鎖法則を適用:

e^{5 c o s (x) l n (x)} \frac{d}{d x} (5 cos (x) ln (x))

= e^{5 c o s (x) l n (x)} \frac{d}{d x} (5 cos (x) ln (x))

\frac{d}{d x} (5 cos (x) ln (x)) = 5 (- sin (x) ln (x) + \frac{cos ( x )}{x})

= e^{5 c o s (x) l n (x)} \cdot 5 (- sin (x) ln (x) + \frac{cos ( x )}{x})

e^{5 c o s (x) l n (x)} = x^{5 c o s (x)}

= 5 x^{5 c o s (x)} (- sin (x) ln (x) + \frac{cos ( x )}{x})