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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

\frac{\partial}{\partial x} (x^{2} y + 5 y^{3})

\frac{d x}{d t} = 3 t^{2} (x^{2} + 4)

\frac{\partial}{\partial x} (9 \frac{x yz}{x + y + z})

\int \frac{3}{x + 3} d x

x^{c o s (x)}

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