What is the derivative of (sin(3x))^{ln(x)} ?

The derivative of (sin(3x))^{ln(x)} is x^{ln(sin(3x))}((ln(sin(3x)))/x+3cot(3x)ln(x))

What is the first derivative of (sin(3x))^{ln(x)} ?

The first derivative of (sin(3x))^{ln(x)} is x^{ln(sin(3x))}((ln(sin(3x)))/x+3cot(3x)ln(x))

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derivative of

(sin (3 x))^{l n (x)}

x^{l n (s i n (3 x))} (\frac{ln ( sin ( 3 x ) )}{x} + 3 cot (3 x) ln (x))

Solution steps

\frac{d}{d x} ((sin (3 x))^{l n (x)})

Apply exponent rule:

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

(sin (3 x))^{l n (x)} = e^{l n (x) l n (s i n (3 x))}

= \frac{d}{d x} (e^{l n (x) l n (s i n (3 x))})

Apply the chain rule:

e^{l n (x) l n (s i n (3 x))} \frac{d}{d x} (ln (x) ln (sin (3 x)))

= e^{l n (x) l n (s i n (3 x))} \frac{d}{d x} (ln (x) ln (sin (3 x)))

\frac{d}{d x} (ln (x) ln (sin (3 x))) = \frac{ln ( sin ( 3 x ) )}{x} + 3 cot (3 x) ln (x)

= e^{l n (x) l n (s i n (3 x))} (\frac{ln ( sin ( 3 x ) )}{x} + 3 cot (3 x) ln (x))

e^{l n (x) l n (s i n (3 x))} = x^{l n (s i n (3 x))}

= x^{l n (s i n (3 x))} (\frac{ln ( sin ( 3 x ) )}{x} + 3 cot (3 x) ln (x))

What is the derivative of (sin(3x))^{ln(x)} ?
The derivative of (sin(3x))^{ln(x)} is x^{ln(sin(3x))}((ln(sin(3x)))/x+3cot(3x)ln(x))
What is the first derivative of (sin(3x))^{ln(x)} ?
The first derivative of (sin(3x))^{ln(x)} is x^{ln(sin(3x))}((ln(sin(3x)))/x+3cot(3x)ln(x))