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

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

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

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

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

3 sin^{3 x} (x) (ln (sin (x)) + x cot (x))

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

= 3 sin^{3 x} (x) (ln (sin (x)) + x cot (x))

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