Derivative of \( \displaystyle \frac{e^{5 x}}{5 x + 1} \)

Problem 2.518 · hard

1. \[ \frac{d}{d x} \frac{e^{5 x}}{5 x + 1} \]
   rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
2. \[ = e^{5 x} \frac{d}{d x} \frac{1}{5 x + 1} + \frac{\frac{d}{d x} e^{5 x}}{5 x + 1} \]
   productApply the product rule.✓ Proved
3. \[ = e^{5 x} \frac{d}{d x} \frac{1}{5 x + 1} + \frac{5 e^{5 x}}{5 x + 1} \]
   derivativeDifferentiate the first term.✓ Proved
4. \[ = \frac{5 e^{5 x}}{5 x + 1} - \frac{e^{5 x} \frac{d}{d x} \left(5 x + 1\right)}{\left(5 x + 1\right)^{2}} \]
   chainApply the chain rule to the second term.✓ Proved
5. \[ = \frac{5 e^{5 x}}{5 x + 1} - \frac{5 e^{5 x}}{\left(5 x + 1\right)^{2}} \]
   derivative algebraDifferentiate the inner function 5*x + 1. Rewrite the terms using positive exponents.✓ Proved
6. \[ = \frac{5 \left(5 x + 1\right) e^{5 x} - 5 e^{5 x}}{\left(5 x + 1\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
7. \[ = \frac{25 x e^{5 x}}{\left(5 x + 1\right)^{2}} \]
   algebra simplifyDistribute the numerator. Simplify the numerator by canceling terms.✓ Proved