Derivative of \( \displaystyle \frac{3 x}{2} - \frac{\sin{\left(6 x + 4 \right)}}{4} \)
Problem 2.2379 · hard Video
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x + 4 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{3 x}{2} - \frac{\sin{\left(6 x + 4 \right)}}{4}\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} \frac{\sin{\left(6 x + 4 \right)}}{4} \]sumApply the sum rule for differentiation.✓ Proved
- \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{\frac{d}{d x} \sin{\left(6 x + 4 \right)}}{4} \]constantPull out the constant factor 1/4.✓ Proved
- \[ = - \frac{\cos{\left(6 x + 4 \right)} \frac{d}{d x} \left(6 x + 4\right)}{4} + \frac{d}{d x} \frac{3 x}{2} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{3 \cos{\left(6 x + 4 \right)}}{2} + \frac{d}{d x} \frac{3 x}{2} \]derivative algebraDifferentiate the inner function 6*x + 4. Simplify the constant product (1/4)*6.✓ Proved
- \[ = \frac{3}{2} - \frac{3 \cos{\left(6 x + 4 \right)}}{2} \]derivative simplifyDifferentiate the remaining term 3*x/2. Factor out the common term 3/2.✓ Proved
Answer \( \frac{3}{2} - \frac{3 \cos{\left(6 x + 4 \right)}}{2} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 |
| 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: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the final result is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.