What is the derivative of (arctan(x)^x) ?

The derivative of (arctan(x)^x) is arctan^x(x)(ln(arctan(x))+x/(arctan(x)(x^2+1)))

What is the first derivative of (arctan(x)^x) ?

The first derivative of (arctan(x)^x) is arctan^x(x)(ln(arctan(x))+x/(arctan(x)(x^2+1)))

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\frac{d}{d x} ((arctan (x))^{x})

arctan^{x} (x) (ln (arctan (x)) + \frac{x}{arctan ( x ) ( x ^{2} + 1 )})

Solution steps

\frac{d}{d x} ((arctan (x))^{x})

Apply exponent rule:

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

(arctan (x))^{x} = e^{x l n (a r c t a n (x))}

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

Apply the chain rule:

e^{x l n (a r c t a n (x))} \frac{d}{d x} (x ln (arctan (x)))

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

\frac{d}{d x} (x ln (arctan (x))) = ln (arctan (x)) + \frac{x}{arctan ( x ) ( x ^{2} + 1 )}

= e^{x l n (a r c t a n (x))} (ln (arctan (x)) + \frac{x}{arctan ( x ) ( x ^{2} + 1 )})

e^{x l n (a r c t a n (x))} = arctan^{x} (x)

= arctan^{x} (x) (ln (arctan (x)) + \frac{x}{arctan ( x ) ( x ^{2} + 1 )})

\int 36 x^{5} d x

\frac{\partial}{\partial x} (4 e^{x} ln (y))

x \to 3 lim (18 - x^{2})

\frac{d}{d x} (arctan (x^{2} y))

\int 11 x ln (x) d x

What is the derivative of (arctan(x)^x) ?
The derivative of (arctan(x)^x) is arctan^x(x)(ln(arctan(x))+x/(arctan(x)(x^2+1)))
What is the first derivative of (arctan(x)^x) ?
The first derivative of (arctan(x)^x) is arctan^x(x)(ln(arctan(x))+x/(arctan(x)(x^2+1)))