Derivative of \( \displaystyle \left(3 x - \frac{12}{5}\right) e^{5 x - 3} \)
Problem 2.1420 · medium
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{12}{5}\right) e^{5 x - 3} \).
- \[ \frac{d}{d x} \left(3 x - \frac{12}{5}\right) e^{5 x - 3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(3 x - \frac{12}{5}\right) \frac{d}{d x} e^{5 x - 3} + e^{5 x - 3} \frac{d}{d x} \left(- \frac{12}{5}\right) + e^{5 x - 3} \frac{d}{d x} 3 x \]productApply the product rule.✓ Proved
- \[ = \left(3 x - \frac{12}{5}\right) \frac{d}{d x} e^{5 x - 3} + e^{5 x - 3} \frac{d}{d x} \left(- \frac{12}{5}\right) + 3 e^{5 x - 3} \]derivativeDifferentiate the first term.✓ Proved
- \[ = \left(3 x - \frac{12}{5}\right) \frac{d}{d x} e^{5 x - 3} + 3 e^{5 x - 3} \]derivative algebraDifferentiate the second term. Simplify the expression.✓ Proved
- \[ = \left(3 x - \frac{12}{5}\right) e^{5 x - 3} \frac{d}{d x} \left(5 x - 3\right) + 3 e^{5 x - 3} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = 5 \left(3 x - \frac{12}{5}\right) e^{5 x - 3} + 3 e^{5 x - 3} \]derivativeDifferentiate the exponent.✓ Proved
- \[ = \left(15 x - 12\right) e^{5 x - 3} + 3 e^{5 x - 3} \]algebraDistribute the 5.✓ Proved
- \[ = \left(15 x - 9\right) e^{5 x - 3} \]algebra simplifyFactor out the common exponential term. Combine like terms.✓ Proved
Answer \( \left(15 x - 9\right) e^{5 x - 3} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 2 applies the product rule to each term of the first factor simultaneously, effectively using two rules in one step. The correct approach is to treat (3*x-12/5) as a single factor and apply the product rule once.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 applies the product rule to each term of the first factor simultaneously, effectively using two rules in one step. The correct approach is to treat (3*x-12/5) as a single factor and apply the product rule once.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 2 applies both the product rule and the derivative of the linear terms in a single step, violating the one‑rule‑per‑step rule. The correct approach would separate the product rule application from the subsequent differentiation of the linear terms.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.