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

The derivative of (sin(x)^{x^3}) is sin^{x^3}(x)(3x^2ln(sin(x))+x^3cot(x))

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

The first derivative of (sin(x)^{x^3}) is sin^{x^3}(x)(3x^2ln(sin(x))+x^3cot(x))

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

sin^{x^{3}} (x) (3 x^{2} ln (sin (x)) + x^{3} cot (x))

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (x^{3} ln (sin (x))) = 3 x^{2} ln (sin (x)) + cot (x) x^{3}

= e^{x^{3} l n (s i n (x))} (3 x^{2} ln (sin (x)) + cot (x) x^{3})

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

= sin^{x^{3}} (x) (3 x^{2} ln (sin (x)) + x^{3} cot (x))

What is the derivative of (sin(x)^{x^3}) ?
The derivative of (sin(x)^{x^3}) is sin^{x^3}(x)(3x^2ln(sin(x))+x^3cot(x))
What is the first derivative of (sin(x)^{x^3}) ?
The first derivative of (sin(x)^{x^3}) is sin^{x^3}(x)(3x^2ln(sin(x))+x^3cot(x))