Derivative of \( \displaystyle \frac{5 x}{5 x + 1} \)
Problem 2.540 · medium
Differentiate \( \displaystyle f(x) = \frac{5 x}{5 x + 1} \).
- \[ \frac{d}{d x} \frac{5 x}{5 x + 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 5 \frac{d}{d x} \frac{x}{5 x + 1} \]constant-multiple rewritePull out the constant factor 5. Rewrite the denominator using a negative exponent.✓ Proved
- \[ = 5 x \frac{d}{d x} \frac{1}{5 x + 1} + \frac{5 \frac{d}{d x} x}{5 x + 1} \]productApply the product rule.✓ Proved
- \[ = 5 x \frac{d}{d x} \frac{1}{5 x + 1} + \frac{5}{5 x + 1} \]derivativeDifferentiate x.✓ Proved
- \[ = - \frac{5 x \frac{d}{d x} \left(5 x + 1\right)}{\left(5 x + 1\right)^{2}} + \frac{5}{5 x + 1} \]chainApply the chain rule to the power term.✓ Proved
- \[ = - \frac{25 x}{\left(5 x + 1\right)^{2}} + \frac{5}{5 x + 1} \]derivative algebra algebraDifferentiate the inner function 5*x + 1. Simplify the negative exponents. Find a common denominator.✓ Proved
- \[ = \frac{5}{\left(5 x + 1\right)^{2}} \]simplify simplifyCombine the fractions. Final simplified form.✓ Proved
Answer \( \frac{5}{\left(5 x + 1\right)^{2}} \)
✓ 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 undefined where 5*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 5*x + 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 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21
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.