Derivative of \( \displaystyle \frac{5 x}{2} - \frac{5 \sin{\left(6 x - 2 \right)}}{12} \)
Problem 2.2271 · hard Video
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x - 2 \right)}}{12} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(6 x - 2 \right)}}{12}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(6 x - 2 \right)}}{12} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x - 2 \right)}}{12} \]constant-multipleFactor out the constant from the second term.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x - 2 \right)}}{12} \]constant-multipleFactor out the constant from the first term.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(6 x - 2 \right)}}{12} \]derivativeDifferentiate x.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x - 2 \right)} \frac{d}{d x} \left(6 x - 2\right)}{12} + \frac{5}{2} \]trigApply the derivative rule for sine.✓ Proved
- \[ = - \frac{5 \cos{\left(6 x - 2 \right)} \frac{d}{d x} \left(-2\right)}{12} - \frac{5 \cos{\left(6 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{5}{2} \]chainApply the chain rule to the argument of the cosine.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(6 x - 2 \right)}}{2} \]derivative algebra simplifyDifferentiate the components of the chain rule. Simplify the expression by removing the zero term. Simplify the coefficients.✓ Proved
Answer \( \frac{5}{2} - \frac{5 \cos{\left(6 x - 2 \right)}}{2} \)
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
| 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 7 applies two rules at once (chain and algebraic distribution) and mislabels the operation as "chain". The derivative of 6*x-2 should be handled by linearity, not a chain rule, and the step should be split into separate algebraic and derivative steps.qwen3.6:27b-mlx: fail (error) — Step 7 is mathematically incorrect; it attempts to apply the chain rule to the argument of the cosine, but the chain rule applies to the outer function (sine) in Step 6, not to the cosine that results from differentiating sine. Furthermore, Step 7 incorrectly expands the derivative of the inner function (6x-2) into a sum of derivatives, which is not how the chain rule is applied in this context.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 7 is mathematically incorrect; it attempts to apply the chain rule to the argument of the cosine, but the chain rule applies to the outer function (sine) in Step 6, not to the cosine that results from differentiating sine. Furthermore, Step 7 incorrectly expands the derivative of the inner function (6x-2) into a sum of derivatives, which is not how the chain rule is applied in this context.gpt-oss:20b: fail (error) 2026-10-17 — Step 7 applies two rules at once (chain and algebraic distribution) and mislabels the operation as "chain". The derivative of 6*x-2 should be handled by linearity, not a chain rule, and the step should be split into separate algebraic and derivative steps.qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 7 is mathematically incorrect. The chain rule states that d/dx[sin(u)] = cos(u) * u'. The step incorrectly applies the product rule to the argument (6x - 2), resulting in two terms (one for 6x and one for -2) multiplied by the outer derivative, which is not how the chain rule works. The derivative of the argument 6x-2 is simply 6, not a sum of derivatives multiplied by the outer function.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — Step 7 is mathematically incorrect. The chain rule states that d/dx sin(u) = cos(u) * u'. The derivative of the inner function u = 6x - 2 is 6, not a sum of derivatives of its terms treated as separate factors outside the chain rule application. The step incorrectly expands the derivative of the inner function into a product of cosines and individual derivatives, violating the structure of the chain rule.gpt-oss:20b: fail (error) 2026-10-14 — Step 7 incorrectly applies the chain rule: it expands the derivative of (6*x-2) into two terms 6*Derivative(x,x) and -2*Derivative(x,x), whereas the derivative of the constant -2 is 0. This violates the rule that each step must apply only one valid transformation.
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 2026-10-17 with SymPy 1.14.0.