Derivative of \( \displaystyle 5 x \ln{\left(4 x + 1 \right)} - 5 x + \frac{5 \ln{\left(4 x + 1 \right)}}{4} \)
Problem 2.1362 · hard
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(4 x + 1 \right)} - 5 x + \frac{5 \ln{\left(4 x + 1 \right)}}{4} \).
- \[ \frac{d}{d x} \left(5 x \ln{\left(4 x + 1 \right)} - 5 x + \frac{5 \ln{\left(4 x + 1 \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 5 x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} - 5 \]constantThe derivative of 5*x is 5.✓ Proved
- \[ = 5 \frac{d}{d x} x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} - 5 \]constant-multipleFactor out the constant 5.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + 5 \ln{\left(4 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} - 5 \]productApply the product rule to the first term.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + 5 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} - 5 \]derivativeThe derivative of x is 1.✓ Proved
- \[ = \frac{20 x}{4 x + 1} + 5 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} - 5 \]chain algebraApply the chain rule to log(4*x + 1). Simplify the expression inside the parentheses.✓ Proved
- \[ = \frac{20 x}{4 x + 1} + 5 \ln{\left(4 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(4 x + 1 \right)}}{4} - 5 \]constant-multipleFactor out 5/4 from the third term.✓ Proved
- \[ = \frac{20 x}{4 x + 1} + 5 \ln{\left(4 x + 1 \right)} - 5 + \frac{5}{4 x + 1} \]chain constant-multipleApply the chain rule to log(4*x + 1) again. Distribute 5 and simplify the constants.✓ Proved
- \[ = 5 \ln{\left(4 x + 1 \right)} - 5 + \frac{20 x + 5}{4 x + 1} \]algebra algebraCombine the fractions with the same denominator. Factor 5 out of the numerator.✓ Proved
- \[ = 5 \ln{\left(4 x + 1 \right)} \]simplify simplifyCancel the common term (4*x + 1). Final simplification.✓ Proved
Answer \( 5 \ln{\left(4 x + 1 \right)} \)
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.
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 undefined where 4*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 1 = 0 |
| 15 | ✓ 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 computes the derivative of 5x (which is 5), not a constant. The label 'constant' is reserved for differentiating a constant term (result 0) or pulling out a constant factor, neither of which describes differentiating 5x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'constant' but computes the derivative of 5x (which is 5), not a constant. The label 'constant' is reserved for differentiating a constant term (result 0) or pulling out a constant factor, neither of which describes differentiating 5x.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 3 incorrectly labels the differentiation of 5*x as 'constant'; the rule 'constant' applies to the derivative of a constant, not a linear term. Additionally, Step 11 applies distribution and simplification simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.