What is the derivative of (ln(x)^{sec(x)}) ?

The derivative of (ln(x)^{sec(x)}) is ln^{sec(x)}(x)(sec(x)tan(x)ln(ln(x))+(sec(x))/(xln(x)))

What is the first derivative of (ln(x)^{sec(x)}) ?

The first derivative of (ln(x)^{sec(x)}) is ln^{sec(x)}(x)(sec(x)tan(x)ln(ln(x))+(sec(x))/(xln(x)))

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

ln^{s e c (x)} (x) (sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )})

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (sec (x) ln (ln (x))) = sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )}

= e^{s e c (x) l n (l n (x))} (sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )})

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

= ln^{s e c (x)} (x) (sec (x) tan (x) ln (ln (x)) + \frac{sec ( x )}{x ln ( x )})

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