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Derivative of \( \displaystyle - \frac{25 x^{2}}{4} - 5 x + \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \ln{\left(5 x + 2 \right)} \)

Problem 2.2556 · hard Video

Differentiate \( \displaystyle f(x) = - \frac{25 x^{2}}{4} - 5 x + \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \ln{\left(5 x + 2 \right)} \).
  1. \[ \frac{d}{d x} \left(- \frac{25 x^{2}}{4} - 5 x + \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \ln{\left(5 x + 2 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \ln{\left(5 x + 2 \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \ln{\left(5 x + 2 \right)} - 5 \]
    constantDifferentiate the linear term.✓ Proved
  4. \[ = \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(\frac{25 x^{2}}{2} + 10 x + 2\right) + \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) - 5 \]
    productApply the product rule to the logarithmic term.✓ Proved
  5. \[ = \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} + \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) - 5 \]
    powerDifferentiate the polynomial part of the product.✓ Proved
  6. \[ = \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) - 5 + \frac{\left(\frac{25 x^{2}}{2} + 10 x + 2\right) \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} \]
    chainApply the chain rule to the natural logarithm.✓ Proved
  7. \[ = \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) - 5 + \frac{5 \left(\frac{25 x^{2}}{2} + 10 x + 2\right)}{5 x + 2} \]
    constantDifferentiate the inner function of the logarithm.✓ Proved
  8. \[ = \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \left(- \frac{25 x^{2}}{4}\right) - 5 + \frac{\frac{125 x^{2}}{2} + 50 x + 10}{5 x + 2} \]
    algebraDistribute the 5 in the numerator of the last term.✓ Proved
  9. \[ = - \frac{25 x}{2} + \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} - 5 + \frac{\frac{125 x^{2}}{2} + 50 x + 10}{5 x + 2} \]
    powerDifferentiate the first term.✓ Proved
  10. \[ = - \frac{25 x}{2} + \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} - 5 + \frac{\frac{25 x \left(5 x + 2\right)}{2} + 25 x + 10}{5 x + 2} \]
    algebraRewrite the numerator to isolate terms divisible by (5x+2).✓ Proved
  11. \[ = \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} - 5 + \frac{25 x + 10}{5 x + 2} \]
    algebra algebraSplit the fraction. Simplify the remaining terms.✓ Proved
  12. \[ = \left(25 x + 10\right) \ln{\left(5 x + 2 \right)} \]
    algebra simplifyCancel the common factor in the fraction. Combine the constant terms.✓ Proved
Answer \( 5 \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} \)
Top: the graph of f = -25x²/4 - 5x + (25x²/2 + 10x + 2)·log(5x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -25x/2 + (25x + 10)·log(5x + 2) - 5 + 5(25x²/2 + 10x + 2)/(5x + 2).

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
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0line 1 is the problem as stated
2✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
3✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
4✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
5✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
6✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
7✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
8✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
9✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
10✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
11✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
12✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
13✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 5*x + 2 = 0
14✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
answer✓ Provedsympy 1.14.0final line against the stated answer: simplify(a - b) reduced to 0
log is undefined for non-positive arguments
answer, a second way✓ Provedsympy 1.14.0SymPy differentiated f directly and got the stated answer

Reviewers

  • gpt-oss:20b: fail (error) — Step 12 incorrectly replaces the sum 25*x/2 + (25*x+10)/(5*x+2) with 5, which is not algebraically equivalent. The correct simplification would keep the terms separate or combine them properly, but the current step introduces an algebraic error.
  • qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications to reach the stated answer. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (6)
  • qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly applies differentiation rules and algebraic simplifications to reach the stated answer. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
  • gpt-oss:20b: fail (error) 2026-11-13 — Step 12 incorrectly replaces the sum 25*x/2 + (25*x+10)/(5*x+2) with 5, which is not algebraically equivalent. The correct simplification would keep the terms separate or combine them properly, but the current step introduces an algebraic error.
  • qwen3.6:27b-mlx: pass 2026-11-12 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.
  • gpt-oss:20b: pass 2026-11-12
  • qwen3.6:27b-mlx: pass 2026-11-12 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications in the later steps are valid and clearly labeled.
  • gpt-oss:20b: pass 2026-11-12

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 2026-11-13 with SymPy 1.14.0.

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