Derivative of \( \displaystyle x \left(1 - \ln{\left(4 x \right)}\right) \)

Problem 2.54 · hard

Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(4 x \right)}\right) \).

\[ \frac{d}{d x} x \left(1 - \ln{\left(4 x \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = x \frac{d}{d x} \left(1 - \ln{\left(4 x \right)}\right) + \left(1 - \ln{\left(4 x \right)}\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = x \frac{d}{d x} \left(1 - \ln{\left(4 x \right)}\right) - \ln{\left(4 x \right)} + 1 \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = x \left(\frac{d}{d x} 1 - \frac{d}{d x} \ln{\left(4 x \right)}\right) - \ln{\left(4 x \right)} + 1 \]
sumDistribute the derivative over the subtraction.✓ Proved
\[ = - x \frac{d}{d x} \ln{\left(4 x \right)} - \ln{\left(4 x \right)} + 1 \]
derivative constant algebraDifferentiate the constant 1. The derivative of 1 is 0. Distribute the negative sign.✓ Proved
\[ = - \ln{\left(4 x \right)} \]
logarithmic algebra algebra simplifyApply the chain rule to the logarithm. Simplify the fraction inside the derivative. Simplify the product of x and 1/x. Combine the constant terms.✓ Proved

Answer \( - \log{\left(4 x \right)} \)

✓ 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
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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, handles the constant derivative, and uses the chain rule for the logarithmic term. Each step isolates a single rule application, and the labels are appropriate.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, handles the constant derivative, and uses the chain rule for the logarithmic term. Each step isolates a single rule application, and the labels are appropriate.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 applies the linearity of the derivative operator (splitting the derivative of a sum/difference) but labels it 'sum'. While 'sum' is in the vocabulary, it typically refers to algebraic addition, whereas this step is a differentiation rule application. More critically, Step 8 labels the differentiation of log(4x) as 'logarithmic', but the note explicitly says 'Apply the chain rule'. The label 'logarithmic' is acceptable for the base rule, but the step combines the chain rule logic (derivative of inner function) with the log rule. However, the primary defect is Step 4: it transforms `Derivative(1 - log(4*x), x)` into `Derivative(1, x) - Derivative(log(4*x), x)`. This is an application of the linearity of differentiation (often grouped under 'sum' or 'derivative' in loose contexts), but the label 'sum' usually implies algebraic manipulation of sums, not the distribution of the derivative operator. A better label would be 'derivative' or 'sum' if 'sum' covers linearity. Let's look closer. The contract says 'sum' is a valid label. Is it wrong? 'Sum' usually means `d/dx(f+g) = d/dx(f) + d/dx(g)`. This is exactly what happened. So Step 4 is fine. Let's re-evaluate Step 8. `Derivative(log(4*x), x)` becomes `1/(4*x)*4`. This step applies the chain rule AND the logarithmic derivative rule. The label is 'logarithmic'. The note says 'Apply the chain rule'. This is a mismatch. The step applies the derivative of log(u) which is u'/u. This involves the chain rule. The label 'logarithmic' is acceptable for the outer function, but the step performs the full differentiation. Is it a defect to label it 'logarithmic' when it also uses the chain rule? The contract says 'Each step changes ONE thing'. Step 8 changes `Derivative(log(4*x), x)` to `1/(4*x)*4`. This is the result of applying the chain rule to the log function. It's a single conceptual step: differentiating a composite log function. The label 'logarithmic' is arguably insufficient because it ignores the chain rule component, or 'chain' is insufficient because it ignores the log component. However, usually, differentiating `log(g(x))` is treated as a single 'logarithmic' differentiation step in many contexts, or 'chain'. But wait, look at Step 3. `Derivative(x, x)` becomes `1`. Label: 'derivative'. Correct. Step 5: `Derivative(1, x)` becomes `0`. Label: 'derivative'. Correct. Step 6: `0 - ...` becomes `-...`. Label: 'constant'. This is algebraic simplification, not a differentiation rule. 'Constant' usually refers to `d/dx(c) = 0` or `d/dx(cf) = c d/dx(f)`. Here, it's just simplifying `0 - A` to `-A`. This is algebra. Label 'constant' is misleading/wrong. It should be 'algebra' or 'simplify'. Let's check Step 6 again. `1 - log(4*x) + x*(0 - Derivative(log(4*x), x))` -> `1 - log(4*x) + x*(-Derivative(log(4*x), x))`. The change is `0 - D` to `-D`. This is arithmetic/algebra. The label is 'constant'. The vocabulary has 'constant' which usually means 'constant rule' for differentiation. This step is not differentiation. It is simplification. Therefore, the label 'constant' is incorrect for an algebraic simplification step. It should be 'algebra' or 'simplify'. Also Step 8: `Derivative(log(4*x), x)` -> `1/(4*x)*4`. This applies the chain rule. The label is 'logarithmic'. While the function is logarithmic, the rule applied is the chain rule combined with the log derivative. If the system distinguishes 'chain' and 'logarithmic', this step is a hybrid. However, often 'logarithmic' implies the standard derivative of log. But `log(4x)` requires chain rule. If I must pick one, 'chain' is the structural rule, 'logarithmic' is the function type. But Step 6 is definitely a labeling error. 'Constant' is not an algebraic simplification label. 'Algebra' or 'Simplify' is. 'Constant' is a differentiation rule. This step does no differentiation. So it's a defect.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the product rule, handles the constant derivative, and uses the chain rule for the logarithmic term. All steps adhere to the single-change constraint and use valid labels from the fixed vocabulary.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 8 is labeled 'logarithmic' but the note explicitly states 'Apply the chain rule', and the step performs the chain rule differentiation of log(4x). The label should be 'chain' or 'logarithmic' without the misleading note, but given the strict vocabulary, 'logarithmic' is acceptable for the rule, yet the note is contradictory/misleading about the rule name used. More critically, Step 4 applies the derivative to a sum/difference, which is technically the 'sum' rule (or linearity), but it also distributes the derivative operator inside. Step 8 applies the derivative of log(u) which is a combination of the logarithmic rule and the chain rule. The label 'logarithmic' is acceptable, but the note says 'Apply the chain rule'. This is a style issue regarding the note's accuracy vs the label. However, looking closer at Step 8: `Derivative(log(4*x), x)` becomes `1/(4*x)*4`. This is the chain rule applied to the logarithm. The label 'logarithmic' is in the vocabulary. Is it a defect? The prompt says 'A label is correct if it names a rule the step legitimately applied.' The step applies the derivative of log, which involves the chain rule. 'Logarithmic' is a valid label for differentiating a log function. The note is just text. Let's re-read the constraints. 'Look for... a note that would teach a student something false'. The note says 'Apply the chain rule to the logarithm'. This is true. The label is 'logarithmic'. This is also true (it's the derivative of a log). Let's look for a real defect. Step 4: `1*(1 - log(4*x)) + x*Derivative(1 - log(4*x), x)` -> `1 - log(4*x) + x*(Derivative(1, x) - Derivative(log(4*x), x))`. This step does two things: 1. Simplifies `1*(...)` to `(...)`. 2. Distributes the derivative over the subtraction inside: `Derivative(A - B) -> Derivative(A) - Derivative(B)`. The label is 'sum'. The 'sum' rule usually refers to `d/dx(f+g) = d/dx(f) + d/dx(g)`. Here we have `d/dx(1 - log(...))`. This is a sum/difference rule. However, the step *also* simplifies the `1*` term from the previous step. Previous step (3): `1*(1 - log(4*x)) + ...` Current step (4): `1 - log(4*x) + ...` So Step 4 applies algebraic simplification (canceling the 1) AND the sum/difference rule for derivatives. The contract says: 'Each step changes ONE thing. A step that applies two rules at once is a defect.' Step 4 changes `1*(1 - log(4*x))` to `1 - log(4*x)` (algebra/simplify) AND changes `Derivative(1 - log(4*x), x)` to `Derivative(1, x) - Derivative(log(4*x), x)` (sum/derivative linearity). This is a violation of the 'one thing per step' rule.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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.