Derivative of \( \displaystyle \left(\frac{4}{5} - x\right) e^{5 x - 3} \)
Problem 2.1421 · medium
Differentiate \( \displaystyle f(x) = \left(\frac{4}{5} - x\right) e^{5 x - 3} \).
- \[ \frac{d}{d x} \left(\frac{4}{5} - x\right) e^{5 x - 3} \]Start with the derivative of the function.✓ Proved
- \[ = \left(\frac{4}{5} - x\right) \frac{d}{d x} e^{5 x - 3} + e^{5 x - 3} \frac{d}{d x} \left(\frac{4}{5} - x\right) \]productApply the product rule.✓ Proved
- \[ = \left(\frac{4}{5} - x\right) e^{5 x - 3} \frac{d}{d x} \left(5 x - 3\right) + \left(\frac{d}{d x} \frac{4}{5} - \frac{d}{d x} x\right) e^{5 x - 3} \]algebraDistribute the derivative across the terms.✓ Proved
- \[ = 5 \left(\frac{4}{5} - x\right) e^{5 x - 3} - e^{5 x - 3} \]derivativeDifferentiate the individual terms.✓ Proved
- \[ = \left(4 - 5 x\right) e^{5 x - 3} - e^{5 x - 3} \]algebraSimplify the expression.✓ Proved
- \[ = \left(3 - 5 x\right) e^{5 x - 3} \]algebra algebra simplifyFactor out the exponential term. Distribute the 5 inside the parentheses. Combine like terms.✓ Proved
Answer \( \left(3 - 5 x\right) e^{5 x - 3} \)
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 |
| 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 (style) — Step 3 incorrectly labels the application of the chain rule as "algebra". The derivative of exp(5*x-3) should be identified as a chain rule step, not an algebraic manipulation.qwen3.6:27b-mlx: fail (error) — Step 3 applies the chain rule to the exponential term and the sum rule to the linear term simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'algebra' is incorrect for a step that performs differentiation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 applies the chain rule to the exponential term and the sum rule to the linear term simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'algebra' is incorrect for a step that performs differentiation.gpt-oss:20b: fail (style) 2026-10-03 — Step 3 incorrectly labels the application of the chain rule as "algebra". The derivative of exp(5*x-3) should be identified as a chain rule step, not an algebraic manipulation.qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 applies the chain rule to the exponential term and the sum rule to the linear term simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'algebra' is incorrect for a step that performs differentiation.gpt-oss:20b: fail (error) 2026-10-03 — Step 3 applies two rules at once: it uses the chain rule to replace Derivative(exp(5*x-3),x) with exp(5*x-3)*Derivative(5*x-3,x) and also uses the sum rule to split Derivative(4/5 - x,x) into Derivative(4/5,x)-Derivative(x,x). The label "algebra" is incorrect for both operations, violating the single‑rule‑per‑step rule.
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.