Home›Calculus 1›Product rule›Problem 2.861

Derivative of \( \displaystyle \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \)

Problem 2.861 · hard Video

Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).

\[ \frac{d}{d x} \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]
Start with the derivative of the function.✓ Proved
\[ = \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) \frac{d}{d x} \frac{e^{4 x + 2}}{8} + \frac{e^{4 x + 2} \frac{d}{d x} \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)}{8} \]
productApply the product rule.✓ Proved
\[ = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2} \frac{d}{d x} \left(4 x + 2\right)}{8} + \frac{e^{4 x + 2} \frac{d}{d x} \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)}{8} \]
constant-multiplePull out the constant 1/8 from the second term's derivative.✓ Proved
\[ = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} + \frac{e^{4 x + 2} \frac{d}{d x} \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)}{8} \]
derivativeDifferentiate the inner function 4*x + 2.✓ Proved
\[ = \frac{\left(- 4 \sin{\left(4 x + 2 \right)} - 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} + \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} \]
trigDifferentiate the sine and cosine terms using the chain rule.✓ Proved
\[ = \frac{\left(- 4 \sin{\left(4 x + 2 \right)} - 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} + \frac{\left(- 4 \sin{\left(4 x + 2 \right)} + 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]
algebraFactor out -4 from the first term.✓ Proved
\[ = \frac{\left(- \sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} + \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} \]
simplifySimplify the fractions by dividing 4/8.✓ Proved
\[ = - e^{4 x + 2} \sin{\left(4 x + 2 \right)} \]
algebra simplify simplifyFactor out the common term exp(4*x + 2)/2. Combine like terms inside the parentheses. Multiply the remaining terms to get the final result.✓ Proved

Answer \( - e^{4 x + 2} \sin{\left(4 x + 2 \right)} \)

Top: the graph of f = (-sin(4x + 2) + cos(4x + 2))·e^(4x + 2)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (-4sin(4x + 2) - 4cos(4x + 2))·e^(4x + 2)/8 + (-sin(4x + 2) + cos(4x + 2))·e^(4x + 2)/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
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: pass
qwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to differentiate the trigonometric terms but labels the step 'trig', which is not a valid label for differentiation in the provided vocabulary (it should be 'chain' or 'derivative'). Additionally, the note explicitly mentions using the chain rule, contradicting the label.

Senior review claude-sonnet-5-5, 2026-10-13: pass — The derivative is mathematically correct at every line, and SymPy proved each step. The labels name rules that were legitimately applied. The note on line 3 ('pull out the constant') is loosely worded, since that step really carries the 1/8 through while applying the exponential chain rule, but that is a style issue at most.

qwen3.6:27b-mlx: dismiss — 'trig' is in the allowed vocabulary and correctly names differentiating sin and cos; applying it to both terms in one step is one change, and the note mentioning the chain rule does not make the label wrong.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — The derivative is mathematically correct at every line, and SymPy proved each step. The labels name rules that were legitimately applied. The note on line 3 ('pull out the constant') is loosely worded, since that step really carries the 1/8 through while applying the exponential chain rule, but that is a style issue at most.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 5 applies the chain rule to differentiate the trigonometric terms but labels the step 'trig', which is not a valid label for differentiation in the provided vocabulary (it should be 'chain' or 'derivative'). Additionally, the note explicitly mentions using the chain rule, contradicting the label.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule, and trigonometric derivatives in a step-by-step manner. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 applies the chain rule to differentiate the sine and cosine terms but labels the step 'trig'. The label 'trig' is not in the allowed vocabulary for differentiation rules (which includes 'chain'), and the step performs differentiation, not just trigonometric simplification.
gpt-oss:20b: pass 2026-09-26

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 2026-09-27 with SymPy 1.14.0.