Home›Calculus 1›Derivatives of logarithms›Problem 2.2073

Derivative of \( \displaystyle - 5 x + \left(5 x - \frac{15}{4}\right) \ln{\left(4 x - 3 \right)} \)

Problem 2.2073 · hard

Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x - \frac{15}{4}\right) \ln{\left(4 x - 3 \right)} \).

\[ \frac{d}{d x} \left(- 5 x + \left(5 x - \frac{15}{4}\right) \ln{\left(4 x - 3 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x - \frac{15}{4}\right) \ln{\left(4 x - 3 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{15 \ln{\left(4 x - 3 \right)}}{4} \]
algebraDistribute the term.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(4 x - 3 \right)} - \frac{15 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + 5 \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{15 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + 5 \ln{\left(4 x - 3 \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- 5 x\right) - \frac{15 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} \]
productApply the product rule to the middle term.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + 5 \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \left(- 5 x\right) - \frac{15 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} \]
derivative algebraDifferentiate x. Distribute the 5.✓ Proved
\[ = \left(5 x - \frac{15}{4}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + 5 \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \left(- 5 x\right) \]
algebraGroup the log derivative terms.✓ Proved
\[ = \left(5 x - \frac{15}{4}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + 5 \ln{\left(4 x - 3 \right)} - 5 \]
derivativeDifferentiate -5*x.✓ Proved
\[ = 5 \ln{\left(4 x - 3 \right)} - 5 + \frac{\left(5 x - \frac{15}{4}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} \]
chainApply the chain rule to the log term.✓ Proved
\[ = 5 \ln{\left(4 x - 3 \right)} - 5 + \frac{4 \left(5 x - \frac{15}{4}\right)}{4 x - 3} \]
derivative algebraDifferentiate 4*x - 3. Simplify the expression.✓ Proved
\[ = 5 \ln{\left(4 x - 3 \right)} - 5 + \frac{20 x - 15}{4 x - 3} \]
algebra algebraMultiply the fraction. Factor out 5 from the numerator.✓ Proved
\[ = 5 \ln{\left(4 x - 3 \right)} \]
simplify simplifyCancel the common term (4*x - 3). Combine the constants.✓ Proved

Answer \( 5 \ln{\left(4 x - 3 \right)} \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

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 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
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
16	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0
17	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
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
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 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate and the logic is sound.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate and the logic is sound.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies differentiation rules step-by-step, adhering to the single-rule-per-step constraint. All labels are appropriate for the transformations performed.

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 2026-10-11 with SymPy 1.14.0.