(\partial)/(\partial x)(x^{cos(x)})

解

\frac{\partial}{\partial x} (x^{c o s (x)})

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

解答ステップ

\frac{\partial}{\partial x} (x^{c o s (x)})

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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