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

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

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

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

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derivative of

f (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 e^{2 x} d x

\frac{\partial}{\partial y} (\frac{x ^{2}}{4 y ^{4}})

\int - \frac{1}{2} tan (2 x) d x

\int \frac{1}{z ^{2} + 4} d z

y = x, y = 12 x, x = 16

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