Derivative of \( \displaystyle x + \left(\frac{3}{4} - x\right) \ln{\left(4 x - 3 \right)} \)
Problem 2.1857 · hard
Differentiate \( \displaystyle f(x) = x + \left(\frac{3}{4} - x\right) \ln{\left(4 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(x + \left(\frac{3}{4} - x\right) \ln{\left(4 x - 3 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(\frac{3}{4} - x\right) \ln{\left(4 x - 3 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(\frac{3}{4} - x\right) \ln{\left(4 x - 3 \right)} + 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = \left(\frac{3}{4} - x\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} \left(\frac{3}{4} - x\right) + 1 \]productApply the product rule to the second term.✓ Proved
- \[ = \left(\frac{3}{4} - x\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} - \ln{\left(4 x - 3 \right)} + 1 \]derivativeThe derivative of 3/4 - x is -1.✓ Proved
- \[ = \frac{\left(\frac{3}{4} - x\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} - \ln{\left(4 x - 3 \right)} + 1 \]logarithmicApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{4 \left(\frac{3}{4} - x\right)}{4 x - 3} - \ln{\left(4 x - 3 \right)} + 1 \]derivative algebraThe derivative of 4*x - 3 is 4. Simplify the expression.✓ Proved
- \[ = \frac{3 - 4 x}{4 x - 3} - \ln{\left(4 x - 3 \right)} + 1 \]algebraDistribute the 4 in the numerator.✓ Proved
- \[ = - \ln{\left(4 x - 3 \right)} \]algebra simplify simplifyNotice that 3 - 4x is -(4x - 3). Simplify the fraction. Combine the constants.✓ Proved
Answer \( - \ln{\left(4 x - 3 \right)} \)
✓ 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 undefined where 4*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (style) 2026-10-08 — Step 3 labels the evaluation of Derivative(x, x) as 'constant', but the derivative of x is 1, not a constant function derivative rule (which applies to d/dx(c)=0). The label should be 'derivative'. Step 6 labels the application of the chain rule for the logarithm as 'logarithmic', but the step explicitly computes the inner derivative, making 'chain' the more precise label for the structural change, or at least 'logarithmic' is acceptable if it implies the chain rule application for logs. However, the primary defect is Step 3's mislabeling.
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 2026-10-08 with SymPy 1.14.0.