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Derivative of \( \displaystyle \frac{\left(5 \sin{\left(2 x - 3 \right)} - 5 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \)

Problem 2.1193 · hard Video

Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).

\[ \frac{d}{d x} \frac{\left(5 \sin{\left(2 x - 3 \right)} - 5 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{5 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{4} + \frac{5 e^{2 x - 3} \frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right)}{4} \]
productApply the product rule.✓ Proved
\[ = \frac{5 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{4} + \frac{5 \left(\frac{d}{d x} \sin{\left(2 x - 3 \right)} - \frac{d}{d x} \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]
sumDistribute the derivative over the subtraction.✓ Proved
\[ = \frac{5 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} + \frac{5 \left(2 \sin{\left(2 x - 3 \right)} + 2 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]
chain algebraApply the chain rule to the sine, cosine, and exponential terms. Simplify the signs.✓ Proved
\[ = 5 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \]
algebra algebra simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved

Answer \( 5 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \)

Top: the graph of f = (5sin(2x - 3) - 5cos(2x - 3))·e^(2x - 3)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (5sin(2x - 3) - 5cos(2x - 3))·e^(2x - 3)/2 + (10sin(2x - 3) + 10cos(2x - 3))·e^(2x - 3)/4.

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. A reviewer's objection to the explanation stands; see the receipt. 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: fail (error) — Step 5 applies the chain rule to multiple terms (sin, cos, and exp) in a single step, violating the rule that each step must change only one thing. This combines several operations at once and should be split into separate steps.
qwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to three distinct terms simultaneously, violating the constraint that each step must change only one thing. It should be split into separate steps for the sine, cosine, and exponential derivatives.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the chain rule to three distinct terms simultaneously, violating the constraint that each step must change only one thing. It should be split into separate steps for the sine, cosine, and exponential derivatives.
gpt-oss:20b: fail (error) 2026-10-03 — Step 5 applies the chain rule to multiple terms (sin, cos, and exp) in a single step, violating the rule that each step must change only one thing. This combines several operations at once and should be split into separate steps.
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint and using valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29

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 2026-10-03 with SymPy 1.14.0.