Derivative of \( \displaystyle \ln{\left(2 x - 3 \right)}^{2 x - 3} \)
Problem 2.510 · hard Video
Differentiate \( \displaystyle f(x) = \ln{\left(2 x - 3 \right)}^{2 x - 3} \).
- \[ \frac{d}{d x} \ln{\left(2 x - 3 \right)}^{2 x - 3} \]rewriteRewrite the function using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} e^{\left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)}} \]chainApply the chain rule to the exponential function.✓ Checked independently
- \[ = e^{\left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)}} \frac{d}{d x} \left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\left(2 x - 3\right) \frac{d}{d x} \ln{\left(\ln{\left(2 x - 3 \right)} \right)} + \ln{\left(\ln{\left(2 x - 3 \right)} \right)} \frac{d}{d x} \left(2 x - 3\right)\right) e^{\left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)}} \]derivativeDifferentiate both parts of the product.✓ Proved
- \[ = \left(\left(2 x - 3\right) \frac{d}{d x} \ln{\left(\ln{\left(2 x - 3 \right)} \right)} + 2 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}\right) e^{\left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)}} \]derivativeDifferentiate the linear term 2*x - 3.✓ Proved
- \[ = \left(\frac{\left(2 x - 3\right) \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{\ln{\left(2 x - 3 \right)}} + 2 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}\right) e^{\left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)}} \]chainApply the chain rule to the nested logarithm.✓ Proved
- \[ = \left(2 \ln{\left(\ln{\left(2 x - 3 \right)} \right)} + \frac{2}{\ln{\left(2 x - 3 \right)}}\right) e^{\left(2 x - 3\right) \ln{\left(\ln{\left(2 x - 3 \right)} \right)}} \]derivative algebraDifferentiate the inner logarithm. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(2 \ln{\left(\ln{\left(2 x - 3 \right)} \right)} + \frac{2}{\ln{\left(2 x - 3 \right)}}\right) \ln{\left(2 x - 3 \right)}^{2 x - 3} \]rewriteConvert the exponential form back to the original base-exponent form.✓ Checked independently
- \[ = 2 \left(\ln{\left(\ln{\left(2 x - 3 \right)} \right)} + \frac{1}{\ln{\left(2 x - 3 \right)}}\right) \ln{\left(2 x - 3 \right)}^{2 x - 3} \]constant-multipleFactor out the constant 2.✓ Proved
Answer \( \left(2 \log{\left(\log{\left(2 x - 3 \right)} \right)} + \frac{2}{\log{\left(2 x - 3 \right)}}\right) \log{\left(2 x - 3 \right)}^{2 x - 3} \)
Lines: 9 proved, 2 checked by independent solution. 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 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left 2*(log(2*x - 3)*log(log(2*x - 3)) + 1)*(-exp((2*x - 3)*log(log(2*x - 3))) + log(2*x - 3)**(2*x - 3))/log(2*x - 3); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: line 2 integrand is exp(e*log(base)), the definition of base**e for base=log(2x-3)>0 region, same variable x ( checks/2.510/line2-b067308a.py)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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*x - 3) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*x - 3) = 0 |
| 9 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left 2*(log(2*x - 3)*log(log(2*x - 3)) + 1)*(exp((2*x - 3)*log(log(2*x - 3))) - log(2*x - 3)**(2*x - 3))/log(2*x - 3); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: exp(e*log(base)) is replaced by base**e (definition of the power), with the common factor unchanged ( checks/2.510/line9-b0819bfb.py)log is undefined for non-positive arguments undefined where log(2*x - 3) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(2*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 log(2*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 (style) — Step 3 incorrectly labels the chain rule as "product" – the derivative of exp(u) is exp(u)*u', not a product rule. Step 4 also mislabels the product rule as "derivative". These labeling errors violate the contract on rule granularity.qwen3.6:27b-mlx: pass — The solution correctly applies logarithmic differentiation by rewriting the function as an exponential, then systematically applies the chain and product rules. Each step isolates a single rule application, and the labels are consistent with the provided vocabulary.
Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct and the final answer matches. Labels on lines 2 to 4 are off: the rewrite into exp is labelled 'chain', the chain-rule step is labelled 'product', and the product-rule step is labelled 'derivative'. A student would learn the wrong rule names.
gpt-oss:20b: uphold — The step into line 3 is the chain rule on exp(u), giving exp(u)*u', yet it is labelled 'product'. The product rule is applied only in the step into line 4, which is labelled 'derivative'. The labels on lines 2 to 4 are each shifted by one step from the work they describe, so there is a real wrong-rule-name defect.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-09 — The math is correct and the final answer matches. Labels on lines 2 to 4 are off: the rewrite into exp is labelled 'chain', the chain-rule step is labelled 'product', and the product-rule step is labelled 'derivative'. A student would learn the wrong rule names.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies logarithmic differentiation by rewriting the function as an exponential, then systematically applies the chain and product rules. Each step isolates a single rule application, and the labels are consistent with the provided vocabulary.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the chain rule as "product" – the derivative of exp(u) is exp(u)*u', not a product rule. Step 4 also mislabels the product rule as "derivative". These labeling errors violate the contract on rule granularity.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 is labeled 'chain' but performs no differentiation; it is merely a rewrite of the function form, which should be labeled 'rewrite'. Step 3 is labeled 'product' but applies the chain rule to the outer exponential function, which is incorrect labeling for the operation performed.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the rule as "product"; the derivative of exp(u) should be obtained by the chain rule, not the product rule. The step itself applies only one rule, but the label is wrong.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies logarithmic differentiation via exponential rewriting. Each step isolates a single rule application (rewrite, chain, product, derivative, algebra) and the labels are consistent with the provided vocabulary.gpt-oss:20b: fail (style) 2026-09-20 — Step 2 incorrectly labels the derivative of exp as "chain"; it should be "derivative". Step 3 applies the product rule to exp(u)*u', which is actually the chain rule; the correct label is "derivative". These mislabelings violate the single‑rule‑per‑step 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.