Derivative of \( \displaystyle - x + \ln{\left(\left(4 x\right)^{x} \right)} \)
Problem 2.2473 · hard Video
Differentiate \( \displaystyle f(x) = - x + \ln{\left(\left(4 x\right)^{x} \right)} \).
- \[ \frac{d}{d x} \left(- x + \ln{\left(\left(4 x\right)^{x} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \ln{\left(\left(4 x\right)^{x} \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} x \ln{\left(4 x \right)} \]rewriteRewrite the log term using the property log(a^b) = b*log(a).✓ Proved
- \[ = \frac{d}{d x} x \ln{\left(4 x \right)} - 1 \]constantThe derivative of -x is -1.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(4 x \right)} + \ln{\left(4 x \right)} \frac{d}{d x} x - 1 \]productApply the product rule to x * log(4*x).✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(4 x \right)} + \ln{\left(4 x \right)} - 1 \]derivativeThe derivative of x is 1.✓ Proved
- \[ = \ln{\left(4 x \right)} \]derivative algebra algebra algebra algebra simplifyApply the chain rule to log(4*x). Simplify the fraction. Simplify the expression inside the parentheses. Combine the constant terms. Distribute the 1. Final simplification.✓ Proved
Answer \( \ln{\left(4 x \right)} \)
✓ 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. 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 |
| 12 | ✓ 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: fail (style) — The solution contains redundant and logically inverted steps at the end (steps 10-12), where the expression is simplified to `log(4*x)` and then immediately expanded back to `log(4*x) + 1 - 1` before simplifying again. This violates the principle of concise, forward-moving derivation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-23 — The solution contains redundant and logically inverted steps at the end (steps 10-12), where the expression is simplified to `log(4*x)` and then immediately expanded back to `log(4*x) + 1 - 1` before simplifying again. This violates the principle of concise, forward-moving derivation.gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: fail (style) 2026-10-22 — Step 7 is labeled 'derivative' but applies the chain rule; the label 'chain' is available in the vocabulary and should be used. Steps 11 and 12 are redundant and logically circular (Step 10 already reached the final answer, Step 11 expands it, and Step 12 collapses it back).gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: fail (style) 2026-10-22 — Steps 11 and 12 are redundant and logically circular; Step 10 already reached the final answer, so expanding it back out and simplifying again is unnecessary and confusing.gpt-oss:20b: pass 2026-10-22
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 2026-10-23 with SymPy 1.14.0.