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Derivative of \( \displaystyle \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \)

Problem 2.344 · hard Video

Differentiate \( \displaystyle f(x) = \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \).
  1. \[ \frac{d}{d x} \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \]
    rewriteRewrite the base and exponent using the exponential and logarithm.✓ Checked independently
  3. \[ = e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \frac{d}{d x} \ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)} \]
    chainApply the chain rule to the exponential function.✓ Proved
  4. \[ = \left(\ln{\left(4 x - 3 \right)} \frac{d}{d x} \cos{\left(4 x - 3 \right)} + \cos{\left(4 x - 3 \right)} \frac{d}{d x} \ln{\left(4 x - 3 \right)}\right) e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \]
    productApply the product rule to the inner expression.✓ Proved
  5. \[ = \left(- 4 \ln{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{4 \cos{\left(4 x - 3 \right)}}{4 x - 3}\right) e^{\ln{\left(4 x - 3 \right)} \cos{\left(4 x - 3 \right)}} \]
    chain algebraDifferentiate the trigonometric and logarithmic terms using the chain rule. Simplify the expression algebraically.✓ Proved
  6. \[ = \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \left(- 4 \ln{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{4 \cos{\left(4 x - 3 \right)}}{4 x - 3}\right) \]
    simplifySubstitute the original function back into the expression.✓ Checked independently
  7. \[ = 4 \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \left(- \ln{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{\cos{\left(4 x - 3 \right)}}{4 x - 3}\right) \]
    simplifyFactor out the common constant 4.✓ Proved
Answer \( \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \left(- 4 \log{\left(4 x - 3 \right)} \sin{\left(4 x - 3 \right)} + \frac{4 \cos{\left(4 x - 3 \right)}}{4 x - 3}\right) \)
Top: the graph of f = (4x - 3)^cos(4x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (4x - 3)^cos(4x - 3)·(-4log(4x - 3)·sin(4x - 3) + 4cos(4x - 3)/(4x - 3)).

✓ Nihil obstat Lines: 7 proved, 2 checked by independent solution. 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
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0line 1 is the problem as stated
2✓ Checked independentlyclaude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0sympy 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
claude-sonnet-5-5 script, run by sympy 1.14.0: for u=4x-3>0, exp(cos(u)*log(u)) = u**cos(u) so the functions (and derivatives) coincide; diff=0 (checks/2.344/line2-e4b7f66e.py)
log is undefined for non-positive arguments
3✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
4✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
5✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 4*x - 3 = 0
6✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 4*x - 3 = 0
7✓ Checked independentlyclaude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0sympy 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
claude-sonnet-5-5 script, run by sympy 1.14.0: replacing exp(cos(u)*log(u)) by u**cos(u) for u=4x-3>0 turns line 6 into line 7; diff=0 (checks/2.344/line7-dc0fb9ad.py)
log is undefined for non-positive arguments
undefined where 4*x - 3 = 0
8✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where 4*x - 3 = 0
answer✓ Provedsympy 1.14.0final 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✓ Provedsympy 1.14.0SymPy 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: pass
  • qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-07: pass — The derivation is mathematically correct: rewrite as exp(cos*log), chain, product, chain, then back-substitution, and the result matches the stated answer. The simplify-labelled steps 7 and 8 are fine. The exp(c*log(u)) = u**c identity holds on the recorded domain 4x-3>0.

  • gpt-oss:20b: dismiss — Step 5 applies a single rule (chain) to evaluate two simple inner derivatives, which the contract counts as one change, and 'chain' is a correct label for it even though the product rule was used in step 4.
Every verdict on record (14)
  • claude-sonnet-5-5: pass 2026-10-07 — The derivation is mathematically correct: rewrite as exp(cos*log), chain, product, chain, then back-substitution, and the result matches the stated answer. The simplify-labelled steps 7 and 8 are fine. The exp(c*log(u)) = u**c identity holds on the recorded domain 4x-3>0.
  • qwen3.6:27b-mlx: pass 2026-09-20
  • gpt-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 2026-09-20
  • gpt-oss:20b: pass 2026-09-20
  • qwen3.6:27b-mlx: pass 2026-09-19
  • deepseek-r1:70b: pass 2026-09-19
  • gpt-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 2026-09-19
  • gpt-oss:20b: pass 2026-09-19
  • qwen3.6:27b-mlx: pass 2026-09-18
  • deepseek-r1:70b: pass 2026-09-18
  • gpt-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.

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