derivative of y^{ln(x})

解

\frac{d}{d x} (y^{l n (x)})

x^{l n (y)} (\frac{ln ( y )}{x} + \frac{ln ( x ) \frac{d y}{d x}}{y})

解答ステップ

\frac{d}{d x} (y^{l n (x)})

指数の規則を適用する:

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

y^{l n (x)} = e^{l n (x) l n (y)}

= \frac{d}{d x} (e^{l n (x) l n (y)})

連鎖法則を適用:

e^{l n (x) l n (y)} \frac{d}{d x} (ln (x) ln (y))

= e^{l n (x) l n (y)} \frac{d}{d x} (ln (x) ln (y))

\frac{d}{d x} (ln (x) ln (y)) = \frac{ln ( y )}{x} + \frac{ln ( x ) \frac{d y}{d x}}{y}

= e^{l n (x) l n (y)} (\frac{ln ( y )}{x} + \frac{ln ( x ) \frac{d y}{d x}}{y})

e^{l n (x) l n (y)} = x^{l n (y)}

= x^{l n (y)} (\frac{ln ( y )}{x} + \frac{ln ( x ) \frac{d y}{d x}}{y})