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

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

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

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

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\frac{d}{d x} ((sin (x))^{c o s (x)})

sin^{c o s (x)} (x) (- sin (x) ln (sin (x)) + cot (x) cos (x))

Solution steps

\frac{d}{d x} ((sin (x))^{c o s (x)})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

= e^{c o s (x) l n (s i n (x))} (- sin (x) ln (sin (x)) + cot (x) cos (x))

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

= sin^{c o s (x)} (x) (- sin (x) ln (sin (x)) + cot (x) cos (x))

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