Derivative of \( \displaystyle \frac{e^{4 x}}{4 x + 1} \)

Problem 2.638 · hard

1. \[ \frac{d}{d x} \frac{e^{4 x}}{4 x + 1} \]
   rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
2. \[ = e^{4 x} \frac{d}{d x} \frac{1}{4 x + 1} + \frac{\frac{d}{d x} e^{4 x}}{4 x + 1} \]
   productApply the product rule.✓ Proved
3. \[ = e^{4 x} \frac{d}{d x} \frac{1}{4 x + 1} + \frac{4 e^{4 x}}{4 x + 1} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \frac{4 e^{4 x}}{4 x + 1} - \frac{e^{4 x} \frac{d}{d x} \left(4 x + 1\right)}{\left(4 x + 1\right)^{2}} \]
   chainApply the chain rule to the second part.✓ Proved
5. \[ = \frac{4 e^{4 x}}{4 x + 1} - \frac{4 e^{4 x}}{\left(4 x + 1\right)^{2}} \]
   derivative algebraDifferentiate the inner function 4*x + 1. Rewrite the negative exponents as denominators.✓ Proved
6. \[ = \frac{4 \left(4 x + 1\right) e^{4 x} - 4 e^{4 x}}{\left(4 x + 1\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
7. \[ = \frac{16 x e^{4 x}}{\left(4 x + 1\right)^{2}} \]
   simplifySimplify the numerator by distributing and canceling terms.✓ Proved