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

The derivative of (tan(1/x)^{sec(1/x)}) is

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

The first derivative of (tan(1/x)^{sec(1/x)}) is

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\frac{d}{d x} ((tan (\frac{1}{x}))^{s e c (\frac{1}{x})})

tan^{s e c (\frac{1}{x})} (\frac{1}{x}) (- \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )})

Solution steps

\frac{d}{d x} ((tan (\frac{1}{x}))^{s e c (\frac{1}{x})})

Apply exponent rule:

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

(tan (\frac{1}{x}))^{s e c (\frac{1}{x})} = e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))}

= \frac{d}{d x} (e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))})

Apply the chain rule:

e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} \frac{d}{d x} (sec (\frac{1}{x}) ln (tan (\frac{1}{x})))

= e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} \frac{d}{d x} (sec (\frac{1}{x}) ln (tan (\frac{1}{x})))

\frac{d}{d x} (sec (\frac{1}{x}) ln (tan (\frac{1}{x}))) = - \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )}

= e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} (- \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )})

e^{s e c (\frac{1}{x}) l n (t a n (\frac{1}{x}))} = tan^{s e c (\frac{1}{x})} (\frac{1}{x})

= tan^{s e c (\frac{1}{x})} (\frac{1}{x}) (- \frac{sec ( \frac{1}{x} ) tan ( \frac{1}{x} ) ln ( tan ( \frac{1}{x} ) )}{x ^{2}} - \frac{sec ^{3} ( \frac{1}{x} )}{x ^{2} tan ( \frac{1}{x} )})

\int_{6}^{10} \frac{1}{( x - 6 ) ^{4}} d x

\int \frac{x}{2 x + 4} d x

x \to 3 lim (\frac{12 - x - 3}{1 + x - x ^{2} + 7})

\frac{d}{d y} (- 2 x y sin (x^{2} y))

\int \frac{1}{sin ^{2} ( θ )} d θ

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