What is the derivative of (sin(4x)^{6x}) ?

The derivative of (sin(4x)^{6x}) is 6sin^{6x}(4x)(ln(sin(4x))+4xcot(4x))

What is the first derivative of (sin(4x)^{6x}) ?

The first derivative of (sin(4x)^{6x}) is 6sin^{6x}(4x)(ln(sin(4x))+4xcot(4x))

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\frac{d}{d x} ((sin (4 x))^{6 x})

6 sin^{6 x} (4 x) (ln (sin (4 x)) + 4 x cot (4 x))

Solution steps

\frac{d}{d x} ((sin (4 x))^{6 x})

Apply exponent rule:

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

(sin (4 x))^{6 x} = e^{6 x l n (s i n (4 x))}

= \frac{d}{d x} (e^{6 x l n (s i n (4 x))})

Apply the chain rule:

e^{6 x l n (s i n (4 x))} \frac{d}{d x} (6 x ln (sin (4 x)))

= e^{6 x l n (s i n (4 x))} \frac{d}{d x} (6 x ln (sin (4 x)))

\frac{d}{d x} (6 x ln (sin (4 x))) = 6 (ln (sin (4 x)) + 4 x cot (4 x))

= e^{6 x l n (s i n (4 x))} \cdot 6 (ln (sin (4 x)) + 4 x cot (4 x))

e^{6 x l n (s i n (4 x))} = sin^{6 x} (4 x)

= 6 sin^{6 x} (4 x) (ln (sin (4 x)) + 4 x cot (4 x))

What is the derivative of (sin(4x)^{6x}) ?
The derivative of (sin(4x)^{6x}) is 6sin^{6x}(4x)(ln(sin(4x))+4xcot(4x))
What is the first derivative of (sin(4x)^{6x}) ?
The first derivative of (sin(4x)^{6x}) is 6sin^{6x}(4x)(ln(sin(4x))+4xcot(4x))