derivative y=(tan(x))^{9/x}

解答

导数

y = (tan (x))^{\frac{9}{x}}

9 tan^{\frac{9}{x}} (x) (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x})

求解步骤

\frac{d}{d x} ((tan (x))^{\frac{9}{x}})

使用指数法则:

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

(tan (x))^{\frac{9}{x}} = e^{\frac{9}{x} l n (t a n (x))}

= \frac{d}{d x} (e^{\frac{9}{x} l n (t a n (x))})

使用连锁律:

e^{\frac{9}{x} l n (t a n (x))} \frac{d}{d x} (\frac{9}{x} ln (tan (x)))

= e^{\frac{9}{x} l n (t a n (x))} \frac{d}{d x} (\frac{9}{x} ln (tan (x)))

\frac{d}{d x} (\frac{9}{x} ln (tan (x))) = 9 (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x})

= e^{\frac{9}{x} l n (t a n (x))} \cdot 9 (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x})

e^{\frac{9}{x} l n (t a n (x))} = tan^{\frac{9}{x}} (x)

= 9 tan^{\frac{9}{x}} (x) (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x})

\int (x^{2} x) d x

d er i v a t i v e 6 x^{\frac{1}{4}}

\int \frac{3 x}{( 1 - x ^{2} ) ^{3}} d x

\frac{\partial}{\partial x} (cos^{2} (x))

d er i v a t i v e (x^{2} (x^{2} + 9 x))^{6}