The power rule and the chains rule

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Homework Statement



How do I know for sure when to use the power rule instead of the chain rule and vice versa?

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The Attempt at a Solution

 
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power rule is used when there are no variables in the raised part (ie. x^2 or 3x^6) but chain rule is used if there are variables in the raised part (ie. d/dx (e^(2x))=e^(2x) * d/dx (2x) = 2e^(2x) Chain rule is used in other situations but this is basically the situation where you would have to decide between chain and power rule as you are asking.
 

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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