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

Problem 2.633 · hard

1. \[ \frac{d}{d x} \left(\sqrt{1 - \left(4 x + 2\right)^{2}} + \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)}\right) \]
   sumStart with the derivative of the entire function.✓ Proved
2. \[ = \frac{d}{d x} \left(4 x + 2\right) \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
   sumSplit the derivative into two parts.✓ Proved
3. \[ = \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} \left(4 x \operatorname{asin}{\left(4 x + 2 \right)} + 2 \operatorname{asin}{\left(4 x + 2 \right)}\right) \]
   algebraExpand the second term.✓ Proved
4. \[ = \frac{d}{d x} 4 x \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} 2 \operatorname{asin}{\left(4 x + 2 \right)} \]
   sumSplit the derivative of the expanded term.✓ Proved
5. \[ = 4 x \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + \operatorname{asin}{\left(4 x + 2 \right)} \frac{d}{d x} 4 x + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} 2 \operatorname{asin}{\left(4 x + 2 \right)} \]
   productApply the product rule to the second term.✓ Proved
6. \[ = 4 x \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{d}{d x} 2 \operatorname{asin}{\left(4 x + 2 \right)} \]
   derivativeDifferentiate 4*x.✓ Proved
7. \[ = 4 x \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + 2 \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} \]
   constant-multipleDifferentiate 2*asin(4*x + 2) by pulling out the constant.✓ Proved
8. \[ = \left(4 x + 2\right) \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} + 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} \]
   algebraCombine the terms involving the derivative of asin.✓ Proved
9. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{\left(4 x + 2\right) \frac{d}{d x} \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
   inverse-trigApply the derivative rule for arcsine.✓ Proved
10. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{4 \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    derivativeDifferentiate 4*x + 2.✓ Proved
11. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 2\right)^{2}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    algebra rewriteMultiply the terms in the last part. Rewrite the square root as a power.✓ Proved
12. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{\frac{d}{d x} \left(1 - \left(4 x + 2\right)^{2}\right)}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    chainApply the chain rule to the power term.✓ Proved
13. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{\frac{d}{d x} 1 - \frac{d}{d x} \left(4 x + 2\right)^{2}}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    sumSplit the derivative inside the power term.✓ Proved
14. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} - \frac{\frac{d}{d x} \left(4 x + 2\right)^{2}}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    derivative constant-multipleDifferentiate the constant 1. Handle the negative sign.✓ Proved
15. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} - \frac{\left(8 x + 4\right) \frac{d}{d x} \left(4 x + 2\right)}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    powerApply the power rule to (4*x + 2)**2.✓ Proved
16. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{- 32 x - 16}{2 \sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    derivative algebraDifferentiate 4*x + 2. Multiply the terms.✓ Proved
17. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} + \frac{- 16 x - 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} + \frac{16 x + 8}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]
    algebraSimplify the first term.✓ Proved
18. \[ = 4 \operatorname{asin}{\left(4 x + 2 \right)} \]
    algebra algebra algebra simplifyCombine the fractions with the same denominator. Expand the numerator. Simplify the numerator. Final simplification.✓ Proved