Derivative of \( \displaystyle \frac{3 \sqrt{\left(5 x + 2\right)^{2} + 1}}{5} \)

Problem 2.1346 · medium

1. \[ \frac{d}{d x} \frac{3 \sqrt{\left(5 x + 2\right)^{2} + 1}}{5} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \sqrt{\left(5 x + 2\right)^{2} + 1}}{5} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(\left(5 x + 2\right)^{2} + 1\right)}{10 \sqrt{\left(5 x + 2\right)^{2} + 1}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = \frac{3 \frac{d}{d x} \left(5 x + 2\right)^{2}}{10 \sqrt{\left(5 x + 2\right)^{2} + 1}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{3 \left(10 x + 4\right) \frac{d}{d x} \left(5 x + 2\right)}{10 \sqrt{\left(5 x + 2\right)^{2} + 1}} \]
   powerApply the power rule to the squared term.✓ Proved
6. \[ = \frac{3 \left(50 x + 20\right)}{10 \sqrt{\left(5 x + 2\right)^{2} + 1}} \]
   derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
7. \[ = \frac{15 x + 6}{\sqrt{\left(5 x + 2\right)^{2} + 1}} \]
   algebraSimplify the expression by canceling 10/10.✓ Proved
8. \[ = \frac{15 x + 6}{\sqrt{25 x^{2} + 20 x + 5}} \]
   simplifyExpand and simplify the denominator.✓ Proved