Derivative of \( \displaystyle \operatorname{asin}{\left(\frac{2 x}{\sqrt{4 x^{2} + 1}} \right)} \)

Problem 2.1453 · hard

1. \[ \frac{d}{d x} \operatorname{asin}{\left(\frac{2 x}{\sqrt{4 x^{2} + 1}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \frac{2 x}{\sqrt{4 x^{2} + 1}}}{\sqrt{- \frac{4 x^{2}}{4 x^{2} + 1} + 1}} \]
   chainApply the chain rule for arcsin.✓ Proved
3. \[ = \frac{- 2 x \frac{d}{d x} \sqrt{4 x^{2} + 1} + \sqrt{4 x^{2} + 1} \frac{d}{d x} 2 x}{\left(4 x^{2} + 1\right) \sqrt{- \frac{4 x^{2}}{4 x^{2} + 1} + 1}} \]
   quotientApply the quotient rule to the inner fraction.✓ Proved
4. \[ = \frac{- 2 x \frac{d}{d x} \sqrt{4 x^{2} + 1} + 2 \sqrt{4 x^{2} + 1}}{\left(4 x^{2} + 1\right) \sqrt{- \frac{4 x^{2}}{4 x^{2} + 1} + 1}} \]
   derivativeDifferentiate the numerator 2*x.✓ Proved
5. \[ = \frac{- \frac{x \frac{d}{d x} \left(4 x^{2} + 1\right)}{\sqrt{4 x^{2} + 1}} + 2 \sqrt{4 x^{2} + 1}}{\left(4 x^{2} + 1\right) \sqrt{- \frac{4 x^{2}}{4 x^{2} + 1} + 1}} \]
   chainApply the chain rule to the square root.✓ Proved
6. \[ = \frac{- \frac{8 x^{2}}{\sqrt{4 x^{2} + 1}} + 2 \sqrt{4 x^{2} + 1}}{\left(4 x^{2} + 1\right) \sqrt{- \frac{4 x^{2}}{4 x^{2} + 1} + 1}} \]
   derivative algebraDifferentiate the polynomial 4*x**2 + 1. Simplify the term 2*x * (8*x / (2*sqrt(4*x**2 + 1))).✓ Proved
7. \[ = \frac{2}{\left(4 x^{2} + 1\right)^{\frac{3}{2}} \sqrt{- \frac{4 x^{2}}{4 x^{2} + 1} + 1}} \]
   algebra algebra algebraFind a common denominator for the numerator. Distribute 2 in the numerator. Simplify the numerator.✓ Proved
8. \[ = \frac{2}{4 x^{2} + 1} \]
   algebra algebra algebra simplifyRewrite the denominator inside the square root. Simplify the fraction inside the square root. Simplify the reciprocal of the square root. Final simplification.✓ Proved