What is the derivative of 9(sin(x))^x ?

The derivative of 9(sin(x))^x is 9sin^x(x)(ln(sin(x))+xcot(x))

What is the first derivative of 9(sin(x))^x ?

The first derivative of 9(sin(x))^x is 9sin^x(x)(ln(sin(x))+xcot(x))

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derivative of

9 (sin (x))^{x}

9 sin^{x} (x) (ln (sin (x)) + x cot (x))

Solution steps

\frac{d}{d x} (9 (sin (x))^{x})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 9 \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))}

= 9 \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)

= 9 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)

= 9 sin^{x} (x) (ln (sin (x)) + x cot (x))

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What is the derivative of 9(sin(x))^x ?
The derivative of 9(sin(x))^x is 9sin^x(x)(ln(sin(x))+xcot(x))
What is the first derivative of 9(sin(x))^x ?
The first derivative of 9(sin(x))^x is 9sin^x(x)(ln(sin(x))+xcot(x))