Derivative of \( \displaystyle \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \)
Problem 2.33 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]constant-multiplePull out the constant factor 1/8.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{8} + \frac{e^{4 x} \frac{d}{d x} \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right)}{8} \]productApply the product rule to the numerator.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) \frac{d}{d x} e^{4 x}}{8} + \frac{\left(\frac{d}{d x} \sin{\left(4 x \right)} - \frac{d}{d x} \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]sumApply the sum rule to the first term.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{2} + \frac{\left(4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) e^{4 x}}{8} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Simplify the signs inside the parentheses.✓ Proved
- \[ = e^{4 x} \sin{\left(4 x \right)} \]algebra algebra simplify simplifyActually, let's just distribute and simplify. Distribute the terms. Combine like terms. Simplify the final expression.✓ Proved
Answer \( e^{4 x} \sin{\left(4 x \right)} \)
✓ 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 |
| 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 (misleading) — Step 5 applies both the chain rule and the derivative of a trigonometric function, but it is labeled only as "trig". The chain rule should also be explicitly named, otherwise a student may think the derivative of sin(4x) is 4cos(4x) without recognizing the chain rule application.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to compute the derivatives of sin(4x), cos(4x), and exp(4x), but is labeled 'trig'. The label 'trig' is not in the allowed vocabulary for differentiation rules (which includes 'chain'), and 'trig' typically refers to trigonometric identities, not differentiation. Furthermore, the step performs three distinct differentiation operations (sin, cos, exp) simultaneously, violating the one-rule-per-step constraint.
Senior review claude-sonnet-5-5, 2026-10-03: pass — Every line is correct and SymPy proved each one equal to the previous line, with the final line matching the stated answer. The 'Actually, let's just distribute' note on line 7 is sloppy wording, and line 8 partly undoes the factoring in line 7, but neither is a mathematical defect.
gpt-oss:20b: dismiss — 'trig' names a rule the step really applied (derivatives of sin and cos), and evaluating the inner derivative of 4x is part of that step, so the missing 'chain' label is not a defect.qwen3.6:27b-mlx: dismiss — 'trig' is in the allowed vocabulary and legitimately applies here. Evaluating all the derivatives with inner function 4x is a single chain-style evaluation step, not independent changes, and SymPy confirms the line is correct.
Every verdict on record (13)
claude-sonnet-5-5: pass 2026-10-03 — Every line is correct and SymPy proved each one equal to the previous line, with the final line matching the stated answer. The 'Actually, let's just distribute' note on line 7 is sloppy wording, and line 8 partly undoes the factoring in line 7, but neither is a mathematical defect.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to compute the derivatives of sin(4x), cos(4x), and exp(4x), but is labeled 'trig'. The label 'trig' is not in the allowed vocabulary for differentiation rules (which includes 'chain'), and 'trig' typically refers to trigonometric identities, not differentiation. Furthermore, the step performs three distinct differentiation operations (sin, cos, exp) simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: fail (misleading) 2026-09-20 — Step 5 applies both the chain rule and the derivative of a trigonometric function, but it is labeled only as "trig". The chain rule should also be explicitly named, otherwise a student may think the derivative of sin(4x) is 4cos(4x) without recognizing the chain rule application.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to differentiate sin(4x) and cos(4x), but labels the step 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules. Additionally, the step performs differentiation (derivative/chain) which is not captured by the label 'trig'.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is labeled 'trig' but performs differentiation using the chain rule; the label 'trig' is incorrect for a differentiation step. Additionally, Step 5 applies the chain rule to differentiate sin(4x), cos(4x), and exp(4x) simultaneously, violating the one-change-per-step rule.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 5 applies the chain rule to differentiate sin(4*x) and cos(4*x) but labels it only as "trig"; it should be labeled "chain" (or both "trig" and "chain" if allowed). Step 7 combines several algebraic manipulations (factoring, distribution, and simplification) in one line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (misleading) 2026-09-19 — Step 5 applies the chain rule to differentiate sin(4x), cos(4x), and exp(4x), but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules; the correct label is 'chain' (or 'derivative' if viewing it as unfolding known derivatives, but 'chain' is more precise for the inner function 4x). Additionally, Step 5 combines the differentiation of three separate terms into one step, violating the 'one thing per step' rule.deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label.gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 5 labels the operation as "trig" but it actually applies the chain rule to differentiate the sine and cosine terms; the rule label does not match the step performed.deepseek-r1:70b: fail 2026-09-17 — Step 7's note incorrectly describes the action as distributing when factoring occurs.
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 2026-09-26 with SymPy 1.14.0.