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

\frac{d}{d x} (\frac{81}{x ^{4}})

y^{^{'}} - \frac{y}{x} = tan (\frac{y}{x})

\int \frac{x}{( x ^{2} + 4 x + 5 ) ^{\frac{3}{2}}} d x

\frac{d y}{d x} = \frac{( x - 4 )}{( y + 5 )}

a x^{2} + b x + c

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))