Home›Calculus 1›Chain rule›Problem 2.109

Derivative of \( \displaystyle \frac{\left(3 \sin{\left(4 x + 1 \right)} - 3 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \)

Problem 2.109 · hard Video

Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \).

\[ \frac{d}{d x} \frac{\left(3 \sin{\left(4 x + 1 \right)} - 3 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
constantStart with the derivative of the function.✓ Proved
\[ = \frac{3 \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{8} + \frac{3 e^{4 x + 1} \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right)}{8} \]
productApply the product rule.✓ Proved
\[ = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{8} + \frac{3 \left(\frac{d}{d x} \sin{\left(4 x + 1 \right)} - \frac{d}{d x} \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
sumDistribute the derivative over the difference.✓ Proved
\[ = \frac{3 \left(4 \sin{\left(4 x + 1 \right)} - 4 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} + \frac{3 \left(4 \sin{\left(4 x + 1 \right)} + 4 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \]
chain algebraApply the chain rule to each term. Simplify the signs.✓ Proved
\[ = 3 e^{4 x + 1} \sin{\left(4 x + 1 \right)} \]
algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved

Answer \( 3 e^{4 x + 1} \sin{\left(4 x + 1 \right)} \)

Top: the graph of f = (3sin(4x + 1) - 3cos(4x + 1))·e^(4x + 1)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (3sin(4x + 1) - 3cos(4x + 1))·e^(4x + 1)/2 + (12sin(4x + 1) + 12cos(4x + 1))·e^(4x + 1)/8.

✓ 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
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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — 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.

Senior review claude-sonnet-5-5, 2026-10-04: pass — With u = 4x+1, the product and chain rules give 3/8·[4(cos u + sin u)e^u + 4(sin u − cos u)e^u] = 3 sin(u)·e^u, which matches the stated answer. Each line is equivalent to the one before, and every rule label names a rule that step actually applied. No panel objections were listed and no weak lines were flagged, so there is nothing to rule on or check.

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-04 — With u = 4x+1, the product and chain rules give 3/8·[4(cos u + sin u)e^u + 4(sin u − cos u)e^u] = 3 sin(u)·e^u, which matches the stated answer. Each line is equivalent to the one before, and every rule label names a rule that step actually applied. No panel objections were listed and no weak lines were flagged, so there is nothing to rule on or check.
qwen3.6:27b-mlx: pass 2026-09-20 — 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.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies the chain rule to three separate derivatives in one line, violating the rule that each step must change only one thing. The step should be split into separate applications for sin, cos, and exp.
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint and using valid labels from the fixed vocabulary.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 5 applies the chain rule to two separate derivatives in a single line, violating the rule that each step must change only one thing. This combines two rule applications into one step.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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.