Derivative of \( \displaystyle - 5 x + \left(5 x + \frac{5}{4}\right) \ln{\left(4 x + 1 \right)} \)
Problem 2.2478 · hard Video
Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x + \frac{5}{4}\right) \ln{\left(4 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 5 x + \left(5 x + \frac{5}{4}\right) \ln{\left(4 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x + \frac{5}{4}\right) \ln{\left(4 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} \]productDistribute the derivative over the product.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} 5 x + \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} \]productApply the product rule to the second term.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + 5 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \frac{5 \ln{\left(4 x + 1 \right)}}{4} \]derivativeDifferentiate 5*x.✓ Proved
- \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 1 \right)} + 5 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{5 \frac{d}{d x} \ln{\left(4 x + 1 \right)}}{4} \]constant-multipleDifferentiate the constant multiple of the third term.✓ Proved
- \[ = \frac{5 x \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} + 5 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{5 \frac{d}{d x} \left(4 x + 1\right)}{4 \left(4 x + 1\right)} \]chainApply the chain rule to the logarithmic terms.✓ Proved
- \[ = \frac{20 x}{4 x + 1} + 5 \ln{\left(4 x + 1 \right)} + \frac{d}{d x} \left(- 5 x\right) + \frac{5}{4 x + 1} \]derivative algebraDifferentiate 4*x + 1. Simplify the products.✓ Proved
- \[ = \frac{20 x}{4 x + 1} + 5 \ln{\left(4 x + 1 \right)} - 5 + \frac{5}{4 x + 1} \]derivativeDifferentiate -5*x.✓ Proved
- \[ = 5 \ln{\left(4 x + 1 \right)} - 5 + \frac{20 x + 5}{4 x + 1} \]algebra algebraCombine the fractions. Factor the numerator.✓ Proved
- \[ = 5 \ln{\left(4 x + 1 \right)} \]simplify algebraSimplify the fraction. Combine the constant terms.✓ 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. 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 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 |
| 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: fail (style) — Step 3 incorrectly labels an algebraic distribution as a product rule. The step only applies the distributive property (an algebra rule), not a product rule. All other steps are fine.qwen3.6:27b-mlx: fail (style) — Step 3 applies the distributive property of the derivative (linearity) to split a product term, but labels it 'product'. The product rule is applied in Step 4. Step 3 should be labeled 'sum' or 'algebra' to reflect the splitting of the sum inside the derivative, not the product rule.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-24 — Step 3 applies the distributive property of the derivative (linearity) to split a product term, but labels it 'product'. The product rule is applied in Step 4. Step 3 should be labeled 'sum' or 'algebra' to reflect the splitting of the sum inside the derivative, not the product rule.gpt-oss:20b: fail (style) 2026-10-24 — Step 3 incorrectly labels an algebraic distribution as a product rule. The step only applies the distributive property (an algebra rule), not a product rule. All other steps are fine.qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 3 applies the product rule to split the term (5*x + 5/4)*log(4*x + 1) into two separate derivative terms. This is an algebraic expansion (distributivity of the derivative over addition), not the product rule. The product rule is applied in Step 4 to the resulting term 5*x*log(4*x + 1). Labeling Step 3 as 'product' is incorrect because it does not apply the product rule formula (u'v + uv'); it merely splits the sum inside the derivative.gpt-oss:20b: fail (style) 2026-10-23 — Step 3 applies an algebraic distribution (splitting (5*x+5/4)*log into 5*x*log + (5/4)*log) but labels it as "product", which is a mislabeling of the rule used.qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 3 incorrectly applies the 'product' rule to split a sum of terms; the sum rule was already applied in Step 2, so splitting (5*x + 5/4)*log(...) into separate derivatives requires the sum rule, not the product rule. Additionally, Step 6 mislabels the application of the constant multiple rule as 'constant-multiple' while simultaneously performing the differentiation of the log term inside, violating the one-rule-per-step constraint or mislabeling the chain rule application that follows.gpt-oss:20b: fail (style) 2026-10-23 — Step 3 incorrectly labels an algebraic distribution as a product rule. The step applies only an algebraic rewrite, not a product rule, and the label should be "algebra" instead of "product".
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.