Derivative of \( \displaystyle x - \ln{\left(\left(3 x\right)^{x} \right)} \)
Problem 2.2413 · hard Video
Differentiate \( \displaystyle f(x) = x - \ln{\left(\left(3 x\right)^{x} \right)} \).
- \[ \frac{d}{d x} \left(x - \ln{\left(\left(3 x\right)^{x} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \ln{\left(\left(3 x\right)^{x} \right)} \]sumApply the difference rule.✓ Proved
- \[ = 1 - \frac{d}{d x} \ln{\left(\left(3 x\right)^{x} \right)} \]constantDifferentiate the constant term.✓ Proved
- \[ = 1 - \frac{d}{d x} x \ln{\left(3 x \right)} \]rewriteUse the property log(a^b) = b*log(a) to simplify the expression before differentiating.✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(3 x \right)} - \ln{\left(3 x \right)} \frac{d}{d x} x + 1 \]productApply the product rule to x * log(3*x).✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(3 x \right)} - \ln{\left(3 x \right)} + 1 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = - \ln{\left(3 x \right)} \]chain algebra algebra algebra simplifyApply the chain rule to log(3*x). Simplify the derivative of the inner function. Simplify the term x * (1/x). Distribute the negative sign. Combine the constant terms.✓ Proved
Answer \( - \ln{\left(3 x \right)} \)
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. How we verify
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 3 is labeled 'constant' but differentiates the variable term x into 1; the rule 'constant' applies to constants, not variables. Additionally, Step 4 applies a logarithmic identity inside the derivative operator without explicitly differentiating, which is a valid algebraic rewrite, but the label 'rewrite' is acceptable; however, the error in Step 3 is a clear mislabeling of the differentiation of x.gpt-oss:20b: fail (style) 2026-10-16 — Step 3 incorrectly labels the differentiation of x as a "constant" rule and the accompanying note claims it is a constant term. The correct label is "derivative" and the note should reflect that x is a variable, not a constant.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 3 is labeled 'constant' but differentiates 'x' (a variable), which is incorrect; it should be labeled 'derivative'. Step 7 is labeled 'chain' but applies the derivative of log and the derivative of the inner function simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.