Derivative of \( \displaystyle \frac{3 \operatorname{asin}{\left(2 x - 1 \right)}}{2} \)

Problem 2.398 · medium

1. \[ \frac{d}{d x} \frac{3 \operatorname{asin}{\left(2 x - 1 \right)}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \operatorname{asin}{\left(2 x - 1 \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(2 x - 1\right)}{2 \sqrt{1 - \left(2 x - 1\right)^{2}}} \]
   chainApply the chain rule to the arcsine function.✓ Proved
4. \[ = \frac{3}{\sqrt{1 - \left(2 x - 1\right)^{2}}} \]
   derivative algebra algebraDifferentiate the inner linear function. Simplify the constant factors. Move the numerator into the square root denominator.✓ Proved
5. \[ = \frac{3}{\sqrt{- 4 x^{2} + 4 x}} \]
   algebra simplify algebraExpand the squared binomial inside the square root. Combine like terms inside the square root. Factor out a 4 from the expression inside the square root.✓ Proved
6. \[ = \frac{3}{2 \sqrt{- x^{2} + x}} \]
   algebraTake the square root of 4 out of the denominator.✓ Proved
7. \[ = \frac{3}{2 \sqrt{x \left(1 - x\right)}} \]
   algebraFactor the quadratic expression inside the square root.✓ Proved