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

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

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

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

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

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

Solution steps

\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))}

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

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

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

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

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