What is the derivative of (sin x)^sin x?

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Homework Statement


f(x)= (sin x)^(sin x)

Homework Equations

The Attempt at a Solution


Taking logarithm on both sides I get:
ln y = ln ((sin x)^(sin x))
 
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Yes...you are on the right track, why stop there? Simplify the expression and differentiate both sides.
 
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lucphysics said:

Homework Statement


f(x)= (sin x)^(sin x)

Homework Equations

The Attempt at a Solution


Taking logarithm on both sides I get:
ln y = ln ((sin x)^(sin x))
That's a good start. Can you rewrite the right side by using a property of the natural log function?
After that, you can differentiate both sides of the resulting equation.
 
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Thank you!
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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