Derivative of \( \displaystyle \left(3 x - \frac{6}{5}\right) e^{5 x - 1} \)

Problem 2.1286 · medium

1. \[ \frac{d}{d x} \left(3 x - \frac{6}{5}\right) e^{5 x - 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(3 x e^{5 x - 1} - \frac{6 e^{5 x - 1}}{5}\right) \]
   algebraDistribute the term (3*x - 6/5) into the exponential.✓ Proved
3. \[ = \frac{d}{d x} 3 x e^{5 x - 1} - \frac{d}{d x} \frac{6 e^{5 x - 1}}{5} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = 3 \frac{d}{d x} x e^{5 x - 1} - \frac{6 \frac{d}{d x} e^{5 x - 1}}{5} \]
   constant-multiplePull out the constant factors from each derivative.✓ Proved
5. \[ = 3 x \frac{d}{d x} e^{5 x - 1} + 3 e^{5 x - 1} \frac{d}{d x} x - \frac{6 \frac{d}{d x} e^{5 x - 1}}{5} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 3 x \frac{d}{d x} e^{5 x - 1} + 3 e^{5 x - 1} - \frac{6 \frac{d}{d x} e^{5 x - 1}}{5} \]
   derivativeDifferentiate x with respect to x.✓ Proved
7. \[ = 15 x e^{5 x - 1} - 3 e^{5 x - 1} \]
   chain algebraApply the chain rule to the exponential term. Simplify the constants and distribute the 3.✓ Proved
8. \[ = \left(15 x - 3\right) e^{5 x - 1} \]
   algebra simplifyFactor out the common exponential term. Combine the constant terms in the parentheses.✓ Proved