Derivative of \( \displaystyle x - \ln{\left(\left(5 x\right)^{x} \right)} \)
Problem 2.2476 · hard Video
Differentiate \( \displaystyle f(x) = x - \ln{\left(\left(5 x\right)^{x} \right)} \).
- \[ \frac{d}{d x} \left(x - \ln{\left(\left(5 x\right)^{x} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \ln{\left(\left(5 x\right)^{x} \right)} \]sumApply the difference rule.✓ Proved
- \[ = 1 - \frac{d}{d x} \ln{\left(\left(5 x\right)^{x} \right)} \]constantDifferentiate the constant term.✓ Proved
- \[ = 1 - \frac{d}{d x} x \ln{\left(5 x \right)} \]rewriteRewrite the logarithm of a power using the property log(a^b) = b*log(a).✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(5 x \right)} - \ln{\left(5 x \right)} \frac{d}{d x} x + 1 \]productApply the product rule to x * log(5*x).✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(5 x \right)} - \ln{\left(5 x \right)} + 1 \]derivative algebraDifferentiate x. Simplify the expression.✓ Proved
- \[ = - \ln{\left(5 x \right)} - \frac{\frac{d}{d x} 5 x}{5} + 1 \]chainApply the chain rule to log(5*x).✓ Proved
- \[ = - \ln{\left(5 x \right)} \]derivative algebra algebra algebra algebraDifferentiate 5*x. Simplify the fraction. Simplify the expression. Distribute the negative sign. Combine the constant terms.✓ Proved
- \[ = - \ln{\left(x \right)} - \ln{\left(5 \right)} \]algebraUse the property log(ab) = log(a) + log(b).✓ Proved
Answer \( - \ln{\left(5 x \right)} \)
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 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 14 | ✓ 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 (error) — Step 3 is labeled 'constant' but differentiates the variable term 'x' into '1'; the rule 'constant' applies only to terms that are constant with respect to the variable of differentiation. Additionally, Step 14 expands the final answer into a sum, which deviates from the stated answer '-log(5*x)' without justification, though the primary defect is the mislabeled differentiation in Step 3.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 3 is labeled 'constant' but differentiates the variable term 'x' into '1'; the rule 'constant' applies only to terms that are constant with respect to the variable of differentiation. Additionally, Step 14 expands the final answer into a sum, which deviates from the stated answer '-log(5*x)' without justification, though the primary defect is the mislabeled differentiation in Step 3.gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 3 is labeled 'constant' but differentiates 'x' (a variable), which is incorrect; it should be labeled 'derivative'. Additionally, Step 3's note claims to differentiate a 'constant term', which is factually false as 'x' is not constant.gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 3 is labeled 'constant' but differentiates the variable term x. The derivative of x is 1, which is a standard derivative rule, not a constant rule. Additionally, Step 3 claims to differentiate a 'constant term' which is factually incorrect as x is not constant.gpt-oss:20b: pass 2026-10-23
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 2026-10-24 with SymPy 1.14.0.