Home›Calculus 1›Derivatives of logarithms›Problem 2.2266

Derivative of \( \displaystyle - 5 x + \left(5 x - \frac{15}{2}\right) \ln{\left(2 x - 3 \right)} \)

Problem 2.2266 · hard Video

Differentiate \( \displaystyle f(x) = - 5 x + \left(5 x - \frac{15}{2}\right) \ln{\left(2 x - 3 \right)} \).

\[ \frac{d}{d x} \left(- 5 x + \left(5 x - \frac{15}{2}\right) \ln{\left(2 x - 3 \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} \left(5 x - \frac{15}{2}\right) \ln{\left(2 x - 3 \right)} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{15 \ln{\left(2 x - 3 \right)}}{2} \]
algebraDistribute the log term.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + \frac{d}{d x} 5 x \ln{\left(2 x - 3 \right)} - \frac{15 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{d}{d x} \left(- 5 x\right) + 5 \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{15 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + 5 \ln{\left(2 x - 3 \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- 5 x\right) - \frac{15 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} \]
productApply the product rule to the middle term.✓ Proved
\[ = 5 x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + 5 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 5 x\right) - \frac{15 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} \]
derivativeDifferentiate x.✓ Proved
\[ = \frac{5 x \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} + 5 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 5 x\right) - \frac{15 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} \]
logarithmicApply the chain rule to the log term.✓ Proved
\[ = \frac{10 x}{2 x - 3} + 5 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 5 x\right) - \frac{15 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} \]
derivative algebra algebraDifferentiate 2*x - 3. Simplify the fraction. Distribute the 5.✓ Proved
\[ = \frac{10 x}{2 x - 3} + 5 \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- 5 x\right) - \frac{15}{2 x - 3} \]
logarithmic algebraDifferentiate the remaining log term. Simplify the constant and fraction.✓ Proved
\[ = \frac{10 x}{2 x - 3} + 5 \ln{\left(2 x - 3 \right)} - 5 - \frac{15}{2 x - 3} \]
derivativeDifferentiate -5*x.✓ Proved
\[ = 5 \ln{\left(2 x - 3 \right)} - 5 + \frac{10 x - 15}{2 x - 3} \]
algebra algebraCombine the fractions. Factor out 5 from the numerator.✓ Proved
\[ = 5 \ln{\left(2 x - 3 \right)} \]
algebra simplifySimplify the fraction. Combine the constants.✓ Proved

Answer \( 5 \ln{\left(2 x - 3 \right)} \)

Top: the graph of f = -5x + (5x - 15/2)·log(2x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5log(2x - 3) - 5 + 2(5x - 15/2)/(2x - 3).

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. The reviewers disagree about how one step is explained; every verdict is in 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	✓ 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 undefined where 2*x - 3 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 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 2*x - 3 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
16	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
17	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
18	✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — Step 8 is labeled 'logarithmic' but applies the chain rule to log(2x-3); the label should be 'chain'. Furthermore, Step 8 applies the chain rule to only one instance of log(2x-3) while leaving another instance undifferentiated in the same step, violating the 'one thing per step' rule.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.