Derivative of \( \displaystyle \frac{5 x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{4} \)
Problem 2.2081 · hard
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{\sin{\left(10 x + 2 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x + 2 \right)}}{4} \]constantFactor out the constant 1/4.✓ Proved
- \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{\frac{d}{d x} \sin{\left(10 x + 2 \right)}}{4} \]constant-multipleFactor out the constant 5/2.✓ Proved
- \[ = \frac{5}{2} - \frac{\frac{d}{d x} \sin{\left(10 x + 2 \right)}}{4} \]derivativeDifferentiate x.✓ Proved
- \[ = - \frac{\cos{\left(10 x + 2 \right)} \frac{d}{d x} \left(10 x + 2\right)}{4} + \frac{5}{2} \]trigApply the chain rule to the sine function.✓ Proved
- \[ = \frac{5}{2} - \frac{5 \cos{\left(10 x + 2 \right)}}{2} \]derivative algebra simplifyDifferentiate the inner function 10*x + 2. Multiply the constants. Simplify the fraction 10/4.✓ Proved
Answer \( \frac{5}{2} - \frac{5 \cos{\left(10 x + 2 \right)}}{2} \)
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.
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 |
| 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: passqwen3.6:27b-mlx: fail (style) — Step 6 is labeled 'trig' but the note explicitly states 'Apply the chain rule', and the step performs the chain rule differentiation of sin(u). The label should be 'chain'. Step 3 is labeled 'constant' but factors out a constant multiplier, which is better described as 'constant-multiple' (or 'algebra' if viewed as rewriting), though 'constant' is sometimes used loosely for pulling out constants, the distinction between step 3 and 4 is muddy; step 4 is clearly 'constant-multiple'. However, the primary defect is the mislabeling of the chain rule step as 'trig'.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (style) 2026-10-11 — Step 6 is labeled 'trig' but the note explicitly states 'Apply the chain rule', and the step performs the chain rule differentiation of sin(u). The label should be 'chain'. Step 3 is labeled 'constant' but factors out a constant multiplier, which is better described as 'constant-multiple' (or 'algebra' if viewed as rewriting), though 'constant' is sometimes used loosely for pulling out constants, the distinction between step 3 and 4 is muddy; step 4 is clearly 'constant-multiple'. However, the primary defect is the mislabeling of the chain rule step as 'trig'.gpt-oss:20b: fail (error) 2026-10-11 — Step 6 applies both the derivative of sin (a trig rule) and the chain rule in one move, violating the one‑rule‑per‑step rule. The correct labeling would require two separate steps: first apply the trig derivative, then apply the chain rule to the inner function.qwen3.6:27b-mlx: fail (style) 2026-10-11 — Step 6 is labeled 'trig' but the note explicitly states 'Apply the chain rule', and the step performs the chain rule differentiation of the sine term. The label should be 'chain'.
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 2026-10-11 with SymPy 1.14.0.