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The derivative from the limit definition

Problem 2.1944 · medium

Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - 4 x^{2} - 2 x + 3 \), then find \( \displaystyle f'(-1) \).
  1. By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
    Reviewed
  2. \[ - 2 h - 2 x - 4 \left(h + x\right)^{2} + 3 = - 4 h^{2} - 8 h x - 2 h - 4 x^{2} - 2 x + 3 \]
    Write out f(x + h).✓ Proved
  3. \[ \frac{- 2 h + 4 x^{2} - 4 \left(h + x\right)^{2}}{h} = - 4 h - 8 x - 2 \]
    Combine and cancel the factor h.✓ Proved
  4. \[ \lim_{h \to 0^+}\left(- 4 h - 8 x - 2\right) = - 8 x - 2 \]
    Now h → 0 by direct substitution.✓ Proved
  5. \[ \left. - 8 x - 2 \right|_{\substack{ x=-1 }} = 6 \]
    At x = -1.✓ Proved
Answer \( f'(x) = - 8 x - 2,\ f'(-1) = 6 \)

Lines: 4 proved, 1 reviewed. 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
LineStatusChecked byDetail
1Reviewedreviewers: gpt-oss:20b, qwen3.6:27b-mlxa sentence; read, not computed
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
4✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
5✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0matches the derivative by the power, quotient and chain rules

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: pass — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result at the specified point.
Every verdict on record (4)
  • gpt-oss:20b: pass 2026-10-09
  • qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result at the specified point.
  • qwen3.6:27b-mlx: fail (error) 2026-10-09 — The algebraic simplification in step 3 is incorrect. The numerator f(x+h) - f(x) simplifies to -4h^2 - 8xh - 2h, which divided by h is -4h - 8x - 2. The solution incorrectly claims this equals -2(2h + 4x + 1) = -4h - 8x - 2, but the subsequent limit evaluation in step 4 treats the expression as if it were -8x - 2 - 4h, leading to the correct final answer by coincidence or error in the intermediate representation, but the explicit equality in step 3 is mathematically false as written (LHS is a rational function, RHS is linear, they are not identical expressions before taking the limit, and the simplification shown is garbled). Specifically, (-2*h + 4*x**2 - 4*(h + x)**2)/h simplifies to -4h - 8x - 2. The RHS given is -4h - 8x - 2. Wait, let's re-read carefully. Step 3 says: (-2*h + 4*x**2 - 4*(h + x)**2)/h = -2*(2*h + 4*x + 1). Let's check the LHS numerator: f(x+h) was calculated in step 2 as -4h^2 - 8hx - 2h - 4x^2 - 2x + 3. f(x) is -4x^2 - 2x + 3. So f(x+h) - f(x) = -4h^2 - 8hx - 2h. Dividing by h gives -4h - 8x - 2. The RHS is -2(2h + 4x + 1) = -4h - 8x - 2. So the equation in step 3 is actually correct algebraically. However, the prompt says the equations are unchecked. I must check the sentences. Step 1 is a standard definition. Step 4 says 'Now h -> 0 by direct substitution'. This is valid for the simplified polynomial. Step 5 is substitution. The solution seems algebraically correct. Why would it fail? Let's look closer at Step 2. It says 'Write out f(x+h)'. The equation is: -2*h - 2*x - 4*(h + x)**2 + 3 = ... This is just expanding f(x+h). It is correct. Step 3 combines f(x+h) - f(x). The LHS of step 3 is (-2*h + 4*x**2 - 4*(h + x)**2)/h. Wait. f(x) = -4x^2 - 2x + 3. f(x+h) = -4(h+x)^2 - 2(h+x) + 3 = -4(h+x)^2 - 2h - 2x + 3. f(x+h) - f(x) = [-4(h+x)^2 - 2h - 2x + 3] - [-4x^2 - 2x + 3] = -4(h+x)^2 - 2h - 2x + 3 + 4x^2 + 2x - 3 = -4(h+x)^2 - 2h + 4x^2. So the numerator is 4x^2 - 2h - 4(h+x)^2. The LHS in step 3 is (-2*h + 4*x**2 - 4*(h + x)**2)/h. This matches. The RHS is -2*(2*h + 4*x + 1) = -4h - 8x - 2. The simplified LHS is -4h - 8x - 2. So the equation is correct. Step 4 takes the limit. Limit of -4h - 8x - 2 as h->0 is -8x - 2. Correct. Step 5 substitutes x=-1. -8(-1) - 2 = 8 - 2 = 6. Correct. The solution is sound. Why did I think it was wrong? I misread the 'unchecked' tag as meaning I should check the algebra, but the instructions say 'do not re-check the algebra of a line marked proved'. However, these lines are marked 'unchecked'. The instructions say: 'Each line ... is either an EQUATION, which a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. The lines here are marked 'unchecked'. This implies I SHOULD check the algebra? Or does 'unchecked' mean the CAS didn't check it? The prompt says 'do not re-check the algebra of a line marked proved'. It does not explicitly say 'check the algebra of a line marked unchecked'. But usually, in these tasks, 'unchecked' means the user wrote it and I need to verify it. If I verify it, it is correct. Is there a sentence error? Step 1: 'By definition...'. Correct. Step 4: 'Now h -> 0 by direct substitution'. This is a valid method for continuous functions (polynomials). Step 5: 'At x = -1'. Correct. The solution appears correct. I will pass it.
  • gpt-oss:20b: pass 2026-10-09

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 generator:structured/derivative_definition, checked 2026-10-09 with SymPy 1.14.0.

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