Derivative of \( \displaystyle \frac{\operatorname{atan}{\left(4 x - 3 \right)}}{4} \)
Problem 2.975 · medium
Differentiate \( \displaystyle f(x) = \frac{\operatorname{atan}{\left(4 x - 3 \right)}}{4} \).
- \[ \frac{d}{d x} \frac{\operatorname{atan}{\left(4 x - 3 \right)}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(4 x - 3 \right)}}{4} \]constant-multiplePull out the constant factor 1/4.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(4 x - 3\right)}{4 \left(\left(4 x - 3\right)^{2} + 1\right)} \]inverse-trigApply the derivative rule for arctan(u).✓ Proved
- \[ = \frac{1}{\left(4 x - 3\right)^{2} + 1} \]derivative simplifyDifferentiate the inner linear function. Cancel the 4 in the numerator and denominator.✓ Proved
- \[ = \frac{1}{16 x^{2} - 24 x + 10} \]algebra simplifyExpand the squared binomial. Combine the constant terms.✓ Proved
Answer \( \frac{1}{\left(4 x - 3\right)^{2} + 1} \)
✓ 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 undefined where 16*x**2 - 24*x + 10 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 - 24*x + 10 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.