Derivative of \( \displaystyle \frac{\sqrt{16 x^{2} + 16 x + 5}}{4} \)

Problem 2.2002 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{16 x^{2} + 16 x + 5}}{4} \]
   constant-multipleStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{16 x^{2} + 16 x + 5}}{4} \]
   constant-multiple rewriteFactor out the constant 1/4. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(16 x^{2} + 16 x + 5\right)}{8 \sqrt{16 x^{2} + 16 x + 5}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{32 x + 16}{8 \sqrt{16 x^{2} + 16 x + 5}} \]
   derivativeDifferentiate the polynomial inside.✓ Proved
5. \[ = \frac{2 \left(2 x + 1\right)}{\sqrt{16 x^{2} + 16 x + 5}} \]
   algebraFactor out 16 from the polynomial.✓ Proved
6. \[ = \frac{4 x + 2}{\sqrt{16 x^{2} + 16 x + 5}} \]
   algebra simplifySimplify the coefficients. Rewrite with a positive exponent and simplify.✓ Proved