What is the derivative of ln((tan(x)^{1/x})) ?

The derivative of ln((tan(x)^{1/x})) is -(ln(tan(x))}{x^2}+\frac{2csc(2x))/x

What is the first derivative of ln((tan(x)^{1/x})) ?

The first derivative of ln((tan(x)^{1/x})) is -(ln(tan(x))}{x^2}+\frac{2csc(2x))/x

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\frac{d}{d x} (ln ((tan (x))^{\frac{1}{x}}))

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

Solution steps

\frac{d}{d x} (ln ((tan (x))^{\frac{1}{x}}))

Apply the chain rule:

\frac{1}{( tan ( x ) ) ^{\frac{1}{x}}} \frac{d}{d x} ((tan (x))^{\frac{1}{x}})

= \frac{1}{( tan ( x ) ) ^{\frac{1}{x}}} \frac{d}{d x} ((tan (x))^{\frac{1}{x}})

\frac{d}{d x} ((tan (x))^{\frac{1}{x}}) = tan^{\frac{1}{x}} (x) (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x})

= \frac{1}{( tan ( x ) ) ^{\frac{1}{x}}} tan^{\frac{1}{x}} (x) (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x})

Simplify

\frac{1}{tan ^{\frac{1}{x}} ( x )} tan^{\frac{1}{x}} (x) (- \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x}) : - \frac{ln ( tan ( x ) )}{x ^{2}} + \frac{2 csc ( 2 x )}{x}

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

x \to 2 lim (5 x^{3} - 7^{2} + 6)

\frac{e ^{x}}{x}

\frac{\partial}{\partial x} (y^{3} - ln (x))

\int 4 x \cdot e^{x} d x

What is the derivative of ln((tan(x)^{1/x})) ?
The derivative of ln((tan(x)^{1/x})) is -(ln(tan(x))}{x^2}+\frac{2csc(2x))/x
What is the first derivative of ln((tan(x)^{1/x})) ?
The first derivative of ln((tan(x)^{1/x})) is -(ln(tan(x))}{x^2}+\frac{2csc(2x))/x