Home›Calculus 1›Logarithmic differentiation›Problem 2.420

Derivative of \( \displaystyle \left(4 x - 2\right)^{4 x - 3} \)

Problem 2.420 · medium Video

Differentiate \( \displaystyle f(x) = \left(4 x - 2\right)^{4 x - 3} \).

\[ \frac{d}{d x} \left(4 x - 2\right)^{4 x - 3} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]
rewriteRewrite the power function using the exponential identity.✓ Checked independently
\[ = e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \frac{d}{d x} \left(4 x - 3\right) \ln{\left(4 x - 2 \right)} \]
chainApply the chain rule to the exponential function.✓ Proved
\[ = \left(\left(4 x - 3\right) \frac{d}{d x} \ln{\left(4 x - 2 \right)} + \ln{\left(4 x - 2 \right)} \frac{d}{d x} \left(4 x - 3\right)\right) e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]
productApply the product rule to the exponent.✓ Proved
\[ = \left(\frac{4 \left(4 x - 3\right)}{4 x - 2} + 4 \ln{\left(4 x - 2 \right)}\right) e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]
derivativeDifferentiate the individual terms in the product.✓ Proved
\[ = \left(4 \ln{\left(4 x - 2 \right)} + \frac{16 x - 12}{4 x - 2}\right) e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]
algebraSimplify the product in the second term.✓ Proved
\[ = \left(4 x - 2\right)^{4 x - 3} \left(4 \ln{\left(4 x - 2 \right)} + \frac{16 x - 12}{4 x - 2}\right) \]
algebra simplify simplify algebraConvert the exponential term back to power form. Factor out the common 4 in the second term. Ensure the fraction is expanded for clarity. Simplify the denominator of the fraction.✓ Checked independently
\[ = \left(4 x - 2\right)^{4 x - 3} \left(4 \ln{\left(4 x - 2 \right)} + \frac{8 x - 6}{2 x - 1}\right) \]
simplifyCancel the common factor of 2.✓ Checked independently

Answer \( \left(4 x - 2\right)^{4 x - 3} \left(4 \log{\left(4 x - 2 \right)} + \frac{16 x - 12}{4 x - 2}\right) \)

Top: the graph of f = (4x - 2)^(4x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (4x - 2)^(4x - 3)·(4log(4x - 2) + (16x - 12)/(4x - 2)).

✓ Nihil obstat Lines: 8 proved, 4 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

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((4x - 2)*(4x - 3)(4x + 2(2x - 1)log(4x - 2) - 3) + (-4x - 2(2x - 1)log(4x - 2) + 3)exp((4x - 3)log(4x - 2)))/(2x - 1); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: difference after doit on positive base 4x-2=p: 0 (`checks/2.420/line2-9e02e57d.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 undefined where 4*x - 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0
7	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: simplify left 2((4x - 2)*(4x - 3)(-4x - 2(2x - 1)log(4x - 2) + 3) + (4x + 2(2x - 1)log(4x - 2) - 3)exp((4x - 3)log(4x - 2)))/(2x - 1); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: exp(elog(b)) = be for b>0; difference: 0 (`checks/2.420/line7-a25a8123.py`) log is undefined for non-positive arguments undefined where 4x - 2 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0
11	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: simplify left (2(2x - 1)(4x - 2)*(4x - 3)(4x + 2(2x - 1)log(4x - 2) - 3) + (4x - 2)(4x - 2)(-4x - 2(2x - 1)log(4x - 2) + 3))/(2x - 1)2; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: with the power factor replaced by P, rational difference: 0 (`checks/2.420/line11-febec48c.py`) log is undefined for non-positive arguments undefined where 4x - 2 = 0 undefined where 2*x - 1 = 0
answer	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: final line against the stated answer: simplify left (-2(2x - 1)(4x - 2)*(4x - 3) + (4x - 2)(4x - 2))(4x + 2(2x - 1)log(4x - 2) - 3)/(2x - 1)2; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: final line vs stated answer with power factor as P, difference: 0 (`checks/2.420/line12-b57884e1.py`) log is undefined for non-positive arguments undefined where 4x - 2 = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-08: pass — The derivative is correct and every line is a valid equivalence. Lines 8 and 9 go back and forth redundantly (9 undoes 8), and line 7 would be better labelled 'rewrite' than 'algebra'. These are style points only.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-08 — The derivative is correct and every line is a valid equivalence. Lines 8 and 9 go back and forth redundantly (9 undoes 8), and line 7 would be better labelled 'rewrite' than 'algebra'. These are style points only.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies logarithmic differentiation, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary. The final simplification steps are algebraically sound and clearly labeled.
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two derivative rules in one line (derivative of 4*x-3 and derivative of log(4*x-2)), violating the one‑rule‑per‑step rule. The label "derivative" is also misleading because it masks the two separate derivative applications.
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, product rule, and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-09-20

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.