What is the derivative of (sin(5x))^{ln(x)} ?

The derivative of (sin(5x))^{ln(x)} is x^{ln(sin(5x))}((ln(sin(5x)))/x+5cot(5x)ln(x))

What is the first derivative of (sin(5x))^{ln(x)} ?

The first derivative of (sin(5x))^{ln(x)} is x^{ln(sin(5x))}((ln(sin(5x)))/x+5cot(5x)ln(x))

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derivative of

(sin (5 x))^{l n (x)}

x^{l n (s i n (5 x))} (\frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x))

Solution steps

\frac{d}{d x} ((sin (5 x))^{l n (x)})

Apply exponent rule:

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

(sin (5 x))^{l n (x)} = e^{l n (x) l n (s i n (5 x))}

= \frac{d}{d x} (e^{l n (x) l n (s i n (5 x))})

Apply the chain rule:

e^{l n (x) l n (s i n (5 x))} \frac{d}{d x} (ln (x) ln (sin (5 x)))

= e^{l n (x) l n (s i n (5 x))} \frac{d}{d x} (ln (x) ln (sin (5 x)))

\frac{d}{d x} (ln (x) ln (sin (5 x))) = \frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x)

= e^{l n (x) l n (s i n (5 x))} (\frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x))

e^{l n (x) l n (s i n (5 x))} = x^{l n (s i n (5 x))}

= x^{l n (s i n (5 x))} (\frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x))

y^{'} = 2 y - 3 y^{2}

\frac{x}{x ^{2} + 81}

\frac{d}{d x} (arctan (\frac{x - 1}{x}))

\int tan (x) sec^{2} (x) d x

\int \frac{1}{2} sin (\frac{2 π t}{5}) d t

What is the derivative of (sin(5x))^{ln(x)} ?
The derivative of (sin(5x))^{ln(x)} is x^{ln(sin(5x))}((ln(sin(5x)))/x+5cot(5x)ln(x))
What is the first derivative of (sin(5x))^{ln(x)} ?
The first derivative of (sin(5x))^{ln(x)} is x^{ln(sin(5x))}((ln(sin(5x)))/x+5cot(5x)ln(x))