Derivative of \( \displaystyle \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \)
Problem 2.344 · hard
Differentiate \( \displaystyle f(x) = \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \).
- \[ \frac{d}{d x} \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \]rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
- \[ = e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \frac{d}{d x} \ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = \left(\ln{\left(4 x - 3 \right)} \frac{d}{d x} \cos{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)} \frac{d}{d x} \ln{\left(4 x - 3 \right)}\right) e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(- 4 \ln{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{4 \cos{\left(4 x - 3 \right)}}{4 x - 3}\right) e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \]chain algebraDifferentiate the trigonometric and logarithmic terms using the chain rule. Simplify the expression algebraically.✓ Proved
- \[ = \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \left(- 4 \ln{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{4 \cos{\left(4 x - 3 \right)}}{4 x - 3}\right) \]simplifySubstitute the original function back into the expression.≈ Checked numerically
- \[ = 4 \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \left(- \ln{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{\cos{\left(4 x - 3 \right)}}{4 x - 3}\right) \]simplifyFactor out the common constant 4.✓ Proved
Answer \( \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \left(- 4 \log{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{4 \cos{\left(4 x - 3 \right)}}{4 x - 3}\right) \)
Lines: 7 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*((4*x - 3)**cos(4*x - 3)*(-(4*x - 3)*log(4*x - 3)*sin(4*x - 3) + cos(4*x - 3)) + ((4*x - 3)*log(4*x - 3)*sin(4*x - 3) - cos(4*x - 3))*exp(log(4*x - 3)*cos(4*x - 3)))/(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*((4*x - 3)**cos(4*x - 3)*((4*x - 3)*log(4*x - 3)*sin(4*x - 3) - cos(4*x - 3)) + (-(4*x - 3)*log(4*x - 3)*sin(4*x - 3) + cos(4*x - 3))*exp(log(4*x - 3)*cos(4*x - 3)))/(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 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 twice (to both cos(4*x-3) and log(4*x-3)) in a single line, violating the rule that each step must change only one thing. The label "chain" is also ambiguous because the product rule was already applied in step 4.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (13)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies the chain rule twice (to both cos(4*x-3) and log(4*x-3)) in a single line, violating the rule that each step must change only one thing. The label "chain" is also ambiguous because the product rule was already applied in step 4.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 applies two chain-rule differentiations at once (for the cosine and the logarithm) but labels only a single "chain" rule. Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule, product rule, and standard derivatives. Each step isolates a single transformation, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 5 applies the chain rule twice (to both the cosine and the logarithm) in a single step, violating the rule that each step must change only one thing. The label "chain" is correct, but the step combines two applications of the chain rule.
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.