Derivative of \( \displaystyle e^{3 x - 7} \)
Problem 2.483 · easy
Differentiate \( \displaystyle f(x) = e^{3 x - 7} \).
- \[ \frac{d}{d x} e^{3 x - 7} \]Start with the derivative of the function.✓ Proved
- \[ = e^{3 x - 7} \frac{d}{d x} \left(3 x - 7\right) \]chainApply the chain rule.✓ Proved
- \[ = e^{3 x - 7} \frac{d}{d x} 3 x - \frac{d}{d x} 7 \]sumDifferentiate the terms in the exponent separately.✓ Proved
- \[ = - e^{3 x - 7} \frac{d}{d x} 7 + 3 e^{3 x - 7} \]productApply the derivative to the first term.✓ Proved
- \[ = - 3 e^{3 x - 7} \frac{d}{d x} 7 + 3 e^{3 x - 7} \]algebraFactor out the common term.✓ Proved
- \[ = 3 e^{3 x - 7} \]derivative simplifyThe derivative of the constant 7 is 0. Simplify the expression.✓ Proved
Answer \( 3 e^{3 x - 7} \)
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 |
| 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 4 incorrectly labels the distribution of the exponential factor as a product rule; it is an algebraic expansion. Step 5 labels the substitution of the derivative of 3*x with 3 as an algebraic step, but it is a derivative application. These mislabelings violate the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly applies the 'sum' rule to split the derivative of a composite function's argument into separate derivative terms without maintaining the chain rule structure, resulting in a syntactically invalid expression (a product minus a derivative). Step 4 further compounds this by misapplying the 'product' rule to a term that is not a product of two functions being differentiated.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 incorrectly applies the 'sum' rule to split the derivative of a composite function's argument into separate derivative terms without maintaining the chain rule structure, resulting in a syntactically invalid expression (a product minus a derivative). Step 4 further compounds this by misapplying the 'product' rule to a term that is not a product of two functions being differentiated.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 incorrectly labels the distribution of the exponential factor as a product rule; it is an algebraic expansion. Step 5 labels the substitution of the derivative of 3*x with 3 as an algebraic step, but it is a derivative application. These mislabelings violate the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the linearity of the derivative operator (splitting the derivative of a sum into a sum of derivatives), but labels it 'sum'. In this vocabulary, 'sum' typically refers to the algebraic sum rule for combining derivatives, but the step is performing the differentiation operation itself on the inner function. More critically, Step 4 is logically flawed: it applies the derivative to the first term of the sum inside the exponent while leaving the derivative operator on the second term, effectively mixing differentiation and algebraic manipulation in a way that doesn't follow a single standard rule cleanly, and the label 'product' is incorrect for differentiating a term inside a sum. The transition from step 2 to 3 should be labeled 'sum' (for the linearity of differentiation) or 'derivative' if viewed as expanding, but step 4 is the real issue: it claims to apply the 'product' rule to `exp(...) * Derivative(3*x - 7, x)`, but it actually differentiates the inner term `3*x` while leaving `Derivative(7, x)` untouched, which is not a standard product rule application. The product rule applies to `u*v`, here we have `u * D(v+w)`. The step breaks `D(3x-7)` into `D(3x) - D(7)`. Step 3 does this split. Step 4 then evaluates `D(3x)` to `3` but leaves `D(7)` as `D(7)`. This is a partial evaluation. The label 'product' is wrong because no product rule is being applied; it's just evaluating one part of a sum. Furthermore, Step 5 factors out `exp(3*x-7)` but the expression in Step 4 is `exp(3*x-7)*3 - exp(3*x-7)*Derivative(7,x)`. Step 5 writes `3*exp(3*x-7) - 3*exp(3*x-7)*Derivative(7,x)`. This is just reordering/multiplication, labeled 'algebra', which is fine. But Step 4 is the problem. It takes `exp(3*x-7) * (Derivative(3*x, x) - Derivative(7, x))` from Step 3 (implicitly, though Step 3 didn't distribute the exp). Wait, Step 3 is `exp(3*x-7) * Derivative(3*x, x) - Derivative(7, x)`. This is ambiguous. Does it mean `exp(...) * (D(3x) - D(7))` or `(exp(...) * D(3x)) - D(7)`? Standard precedence implies the latter, which is wrong because the `exp` should multiply the whole derivative. If it means the former, the notation is missing parentheses. If it means the latter, it's mathematically wrong. Given the next step multiplies the second term by `exp`, it likely meant the former but wrote it ambiguously or incorrectly. Let's look closer. Step 3: `exp(3*x - 7) * Derivative(3*x, x) - Derivative(7, x)`. This is missing parentheses around the derivative terms if the exp is meant to distribute. If it's `(exp * D(3x)) - D(7)`, that's wrong. If it's `exp * (D(3x) - D(7))`, the parentheses are missing. Step 4: `exp(3*x - 7) * 3 - exp(3*x - 7) * Derivative(7, x)`. This step distributes the `exp` term into the subtraction from Step 3 (assuming Step 3 was `exp * (D(3x) - D(7))`). So Step 4 is distributing `exp` over the sum/difference. That is an algebraic step (distributive property). But it is labeled 'product'. And it also evaluates `Derivative(3*x, x)` to `3`. So Step 4 does two things: distributes the exponential factor AND evaluates the derivative of 3x. This violates the 'one thing per step' rule. Also, the label 'product' is wrong for distribution.gpt-oss:20b: fail (error) 2026-09-20 — Step 4 incorrectly labels a distribution as a product rule, and step 5 multiplies the second term by 3, which is not justified. Both steps apply more than one rule or an incorrect rule, violating the contract.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the linearity of the derivative operator to split the derivative of a sum, but labels it 'sum'. While 'sum' is in the vocabulary, the step also implicitly distributes the outer multiplication in Step 4, but Step 3 itself is just linearity. However, the real error is in Step 4: it claims to apply the 'product' rule, but it is actually distributing the derivative operator over the sum from Step 3 (linearity) and evaluating the first term. The label 'product' is incorrect because no product rule (d(uv)) is being applied; rather, the derivative is being applied to terms inside a sum. Furthermore, Step 3's note says 'Differentiate the terms... separately' which implies the derivative operator is being distributed, but the label 'sum' usually refers to the sum rule for derivatives d(u+v)=d(u)+d(v). The transition from 2 to 3 is d(u+v) -> d(u) + d(v). This is the sum rule. But Step 4 is the problem. Step 4 goes from `exp(...) * (D(3x) - D(7))` to `exp(...) * 3 - exp(...) * D(7)`. This is algebraic distribution, not the product rule. The label 'product' is wrong. Also, Step 3 label 'sum' is acceptable for the sum rule of derivatives. But Step 4 label 'product' is definitely wrong as it's just algebraic distribution. Let's look closer. Step 3: `exp(...) * D(3x-7)` -> `exp(...) * (D(3x) - D(7))`. This applies the sum/difference rule to the inner derivative. Label 'sum' is okay. Step 4: `exp(...) * (D(3x) - D(7))` -> `exp(...) * 3 - exp(...) * D(7)`. This is distributing the multiplication. The label is 'product'. The product rule is d(fg) = f'd + gd'. This step is not that. It is algebraic simplification/distribution. So 'product' is a wrong label. Is it a defect? Yes, naming a rule that was not applied. Also, Step 5 is weird. It factors out? No, it just writes `3*exp(...) - 3*exp(...)*D(7)`. Wait, Step 4 result is `exp(...) * 3 - exp(...) * D(7)`. Step 5 is `3*exp(...) - 3*exp(...) * D(7)`. This is just rewriting `exp*3` as `3*exp`. That's algebra. But the note says 'Factor out the common term'. It didn't factor out. It just commuted. And it introduced a 3 in the second term? No, `D(7)` is still there. Wait, look at Step 4 to 5. Step 4: `exp(3*x - 7) * 3 - exp(3*x - 7) * Derivative(7, x)`. Step 5: `3*exp(3*x - 7) - 3*exp(3*x - 7) * Derivative(7, x)`. Where did the extra 3 in the second term come from? It's not there in Step 4. Step 4 has `exp(...) * D(7)`. Step 5 has `3*exp(...) * D(7)`. This is an algebraic error in the step itself, unless `D(7)` was somehow multiplied by 3? No. The step is mathematically incorrect. `A*3 - A*B` became `3*A - 3*A*B`. This is not equal. So it's an error.gpt-oss:20b: fail (style) 2026-09-20 — Step 4 incorrectly labels the operation as a product rule; it merely distributes the exponential over the subtraction, which is an algebraic simplification. The correct label should be "algebra" or "distribute".
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 2026-09-26 with SymPy 1.14.0.