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

The derivative of (sin(x)^x) is sin^x(x)(ln(sin(x))+xcot(x))

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

The first derivative of (sin(x)^x) is sin^x(x)(ln(sin(x))+xcot(x))

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

sin^{x} (x) (ln (sin (x)) + x cot (x))

Solution steps

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

Apply exponent rule:

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

(sin (x))^{x} = e^{x l n (s i n (x))}

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

Apply the chain rule:

e^{x l n (s i n (x))} \frac{d}{d x} (x ln (sin (x)))

= e^{x l n (s i n (x))} \frac{d}{d x} (x ln (sin (x)))

\frac{d}{d x} (x ln (sin (x))) = ln (sin (x)) + x cot (x)

= e^{x l n (s i n (x))} (ln (sin (x)) + x cot (x))

e^{x l n (s i n (x))} = sin^{x} (x)

= sin^{x} (x) (ln (sin (x)) + x cot (x))

\int 5 x ln (x) d x

x \to 8 + lim (ln (x - 8))

x \to 5 lim (4 x + 5)

y = - t^{3}, (2, - 8)

x^{2} + x - 2, - 1, 3

What is the derivative of (sin(x)^x) ?
The derivative of (sin(x)^x) is sin^x(x)(ln(sin(x))+xcot(x))
What is the first derivative of (sin(x)^x) ?
The first derivative of (sin(x)^x) is sin^x(x)(ln(sin(x))+xcot(x))