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

The derivative of (tan(x)^{tan(x)}) is tan^{tan(x)}(x)(sec^2(x)ln(tan(x))+sec^2(x))

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

The first derivative of (tan(x)^{tan(x)}) is tan^{tan(x)}(x)(sec^2(x)ln(tan(x))+sec^2(x))

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\frac{d}{d x} ((tan (x))^{t a n (x)})

tan^{t a n (x)} (x) (sec^{2} (x) ln (tan (x)) + sec^{2} (x))

Solution steps

\frac{d}{d x} ((tan (x))^{t a n (x)})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (tan (x) ln (tan (x))) = sec^{2} (x) ln (tan (x)) + sec^{2} (x)

= e^{t a n (x) l n (t a n (x))} (sec^{2} (x) ln (tan (x)) + sec^{2} (x))

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

= tan^{t a n (x)} (x) (sec^{2} (x) ln (tan (x)) + sec^{2} (x))

What is the derivative of (tan(x)^{tan(x)}) ?
The derivative of (tan(x)^{tan(x)}) is tan^{tan(x)}(x)(sec^2(x)ln(tan(x))+sec^2(x))
What is the first derivative of (tan(x)^{tan(x)}) ?
The first derivative of (tan(x)^{tan(x)}) is tan^{tan(x)}(x)(sec^2(x)ln(tan(x))+sec^2(x))