Derivative of \( \displaystyle \operatorname{asin}{\left(2 x + 1 \right)} \)

Problem 2.1678 · medium

1. \[ \frac{d}{d x} \operatorname{asin}{\left(2 x + 1 \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(2 x + 1\right)}{\sqrt{1 - \left(2 x + 1\right)^{2}}} \]
   chainApply the chain rule for the arcsine function.✓ Proved
3. \[ = \frac{2}{\sqrt{1 - \left(2 x + 1\right)^{2}}} \]
   derivative algebraDifferentiate the inner function 2*x + 1. Multiply the terms.✓ Proved
4. \[ = \frac{2}{\sqrt{- 4 x^{2} - 4 x}} \]
   algebra simplifyExpand the square inside the square root. Simplify the expression inside the square root.✓ Proved
5. \[ = \frac{1}{\sqrt{- x^{2} - x}} \]
   algebra simplifyFactor out 4 from the expression inside the square root. Simplify the fraction.✓ Proved