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

Problem 2.675 · medium

1. \[ \frac{d}{d x} \frac{\sqrt{\left(5 x - 3\right)^{2} + 1}}{5} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{\left(5 x - 3\right)^{2} + 1}}{5} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\left(5 x - 3\right)^{2} + 1\right)}{10 \sqrt{\left(5 x - 3\right)^{2} + 1}} \]
   chain algebraApply the chain rule. Simplify the coefficient.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x - 3\right)^{2}}{10 \sqrt{\left(5 x - 3\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parenthesis.✓ Proved
5. \[ = \frac{\frac{d}{d x} \left(5 x - 3\right)^{2}}{10 \sqrt{\left(5 x - 3\right)^{2} + 1}} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = \frac{\left(10 x - 6\right) \frac{d}{d x} \left(5 x - 3\right)}{10 \sqrt{\left(5 x - 3\right)^{2} + 1}} \]
   powerApply the power rule to (5*x - 3)**2.✓ Proved
7. \[ = \frac{50 x - 30}{10 \sqrt{\left(5 x - 3\right)^{2} + 1}} \]
   chain algebraDifferentiate the inner linear function. Multiply the constants.✓ Proved
8. \[ = \frac{5 x - 3}{\sqrt{\left(5 x - 3\right)^{2} + 1}} \]
   algebraCancel the 10 and 1/10.✓ Proved
9. \[ = \frac{5 x - 3}{\sqrt{25 x^{2} - 30 x + 10}} \]
   simplifyExpand and simplify the radicand.✓ Proved