Derivative of \( \displaystyle 2 x + \ln{\left(x^{2} \right)} + 6 \)
Problem 2.351 · hard Beautiful
Differentiate \( \displaystyle f(x) = 2 x + \ln{\left(x^{2} \right)} + 6 \).
- \[ \frac{d}{d x} \left(2 x + \ln{\left(x^{2} \right)} + 6\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 6 + \frac{d}{d x} 2 x + \frac{d}{d x} \ln{\left(x^{2} \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x + \frac{d}{d x} \ln{\left(x^{2} \right)} \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(x^{2} \right)} + 2 \]constant-multipleThe derivative of 2*x is 2.✓ Proved
- \[ = 2 + \frac{\frac{d}{d x} x^{2}}{x^{2}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 2 + \frac{2}{x} \]power algebraDifferentiate x**2 using the power rule. Simplify the expression.✓ Proved
- \[ = \frac{2 x + 2}{x} \]simplifyCombine into a single fraction.✓ Proved
Answer \( 2 + \frac{2}{x} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 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 1 has no rule label; every step must name a rule from the allowed vocabulary (e.g., "derivative").deepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'constant-multiple' but performs the differentiation of 2*x (which requires the derivative rule for x and constant multiple), not just extracting a constant. Step 8 is unnecessary as the stated answer is 2 + 2/x, not the combined fraction.
Every verdict on record (13)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 is labeled 'constant-multiple' but performs the differentiation of 2*x (which requires the derivative rule for x and constant multiple), not just extracting a constant. Step 8 is unnecessary as the stated answer is 2 + 2/x, not the combined fraction.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 has no rule label; every step must name a rule from the allowed vocabulary (e.g., "derivative").qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 is labeled 'constant-multiple' but performs the differentiation of 2*x into 2, which is a 'derivative' step (specifically applying the constant multiple rule to the derivative of x). The label should be 'derivative' or 'constant-multiple' applied to a derivative step, but since the step computes the derivative, 'derivative' is the primary action, or if viewing it as applying the constant multiple rule to the result of d/dx(x), the label is acceptable but the note is misleadingly specific about the result rather than the rule. More critically, Step 4 combines the evaluation of the derivative of 2*x with the removal of the Derivative wrapper. The rule 'constant-multiple' usually refers to pulling a constant out, e.g., d/dx(c*f) = c*d/dx(f). Here, the step evaluates d/dx(2*x) to 2. This is a 'derivative' step (using power/constant rules). Labeling it 'constant-multiple' is imprecise if it implies just pulling out the 2, but the step actually computes the derivative. However, the biggest issue is Step 4's label vs action. It computes the derivative. The label 'constant-multiple' is often used for d/dx(c*x) = c*d/dx(x). If the step was 2*Derivative(x,x), then 'constant-multiple' would be perfect. But it jumps to 2. This is a 'derivative' step. Also, Step 8 changes the form to (2*x+2)/x which is not the stated answer 2 + 2/x, though algebraically equal. The task was to differentiate, and the stated answer is 2 + 2/x. Step 8 produces a different form. This is a style defect because the solution goes beyond the stated answer form unnecessarily, but more importantly, Step 4's label is questionable. Let's look closer. Step 4: `2 + Derivative(log(x**2), x)`. The previous term was `Derivative(2*x, x)`. The step replaced `Derivative(2*x, x)` with `2`. This is computing a derivative. The rule is 'derivative'. The label 'constant-multiple' is incorrect because 'constant-multiple' is a rule for handling constants in differentiation (d/dx(cf) = c d/dx f), not for evaluating the final derivative of a linear term directly without showing the inner derivative. It's a 'derivative' step.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 4 is labeled 'constant-multiple' but performs the differentiation of 2*x to 2, which is a 'derivative' step (or 'constant-multiple' applied to a derivative that hasn't been taken yet, but the label implies the rule applied in this specific transition, which is evaluating the derivative). More critically, Step 4 combines the evaluation of the derivative of 2x with the simplification of the previous line, but the primary issue is that Step 4's label 'constant-multiple' is incorrect for the transition from Derivative(2*x, x) to 2; that transition is the application of the power rule/derivative rule. The label 'constant-multiple' usually refers to pulling a constant out, e.g., Derivative(2*x, x) -> 2*Derivative(x, x). Here, the step jumps to the final value 2. While 'derivative' would be a better label, the contract allows 'constant-multiple' if it's part of the process, but strictly speaking, the step computes the derivative. However, looking closer at the contract: 'derivative' means unfolding d/dx on a known form. Step 4 unfolds d/dx(2x) to 2. This is a derivative step. Labeling it 'constant-multiple' is a mislabeling of the rule applied. The rule applied is the derivative of a linear term. 'Constant-multiple' is a rule for moving constants outside the derivative operator. The step did not just move a constant; it evaluated the derivative. Therefore, the label is wrong.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 4 is labeled 'constant-multiple' but the note claims it computes the derivative of 2*x; the step actually applies the derivative rule for x (or power rule) and the constant multiple rule simultaneously, violating the one-rule-per-step constraint. Additionally, the final step combines terms into a single fraction which was not requested and changes the form from the stated answer.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-18 — Step 4 applies two rules at once: it uses the constant-multiple rule to pull out the 2 and the derivative rule to differentiate x. These should be separate steps.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18
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.