Derivative of \( \displaystyle \sqrt{x - 3} e^{x - 3} \)
Problem 2.412 · hard
Differentiate \( \displaystyle f(x) = \sqrt{x - 3} e^{x - 3} \).
- \[ \frac{d}{d x} \sqrt{x - 3} e^{x - 3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \sqrt{x - 3} \frac{d}{d x} e^{x - 3} + e^{x - 3} \frac{d}{d x} \sqrt{x - 3} \]product rewriteApply the product rule. Rewrite the square root as a fractional power.✓ Proved
- \[ = \sqrt{x - 3} \frac{d}{d x} e^{x - 3} + \frac{e^{x - 3}}{2 \sqrt{x - 3}} \]powerDifferentiate the power term.✓ Proved
- \[ = \sqrt{x - 3} e^{x - 3} \frac{d}{d x} \left(x - 3\right) + \frac{e^{x - 3}}{2 \sqrt{x - 3}} \]exponentialDifferentiate the exponential term using the chain rule.✓ Proved
- \[ = \sqrt{x - 3} e^{x - 3} + \frac{e^{x - 3}}{2 \sqrt{x - 3}} \]derivative constantDifferentiate the inner function (x - 3). Simplify the derivative of the inner function.✓ Proved
- \[ = \left(\sqrt{x - 3} + \frac{1}{2 \sqrt{x - 3}}\right) e^{x - 3} \]algebra algebraFactor out the common exponential term. Prepare the second term for a common denominator.✓ Proved
- \[ = \left(\frac{2 x - 6}{2 \sqrt{x - 3}} + \frac{1}{2 \sqrt{x - 3}}\right) e^{x - 3} \]algebraMultiply the numerator and denominator by 2 * (x - 3)**(1/2) is not quite right, let's use the previous step's logic: rewrite (x-3)**(1/2) as (x-3)/(x-3)**(1/2).✓ Proved
- \[ = \frac{\left(2 x - 5\right) e^{x - 3}}{2 \sqrt{x - 3}} \]algebra simplifyCombine the terms over a common denominator. Simplify the numerator and the final expression.✓ Proved
Answer \( \frac{\left(x - \frac{5}{2}\right) e^{x - 3}}{\sqrt{x - 3}} \)
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.
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 4 differentiates (x-3)^(1/2) using the power rule but omits the derivative of the inner function (x-3). The correct derivative should include a factor of 1, i.e., (1/2)(x-3)^(-1/2).qwen3.6:27b-mlx: fail (style) — Step 5 is labeled 'exponential' but the note explicitly claims to use the chain rule; while the label is acceptable under the rules, the note is misleading because the chain rule application (differentiating the inner function) is actually performed in Step 6. Additionally, Step 10's note is confusing and self-contradictory regarding the algebraic manipulation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 is labeled 'exponential' but the note explicitly claims to use the chain rule; while the label is acceptable under the rules, the note is misleading because the chain rule application (differentiating the inner function) is actually performed in Step 6. Additionally, Step 10's note is confusing and self-contradictory regarding the algebraic manipulation.gpt-oss:20b: fail (error) 2026-09-21 — Step 4 differentiates (x-3)^(1/2) using the power rule but omits the derivative of the inner function (x-3). The correct derivative should include a factor of 1, i.e., (1/2)(x-3)^(-1/2).qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 5 is labeled 'exponential' but the note explicitly states 'using the chain rule', and the step itself applies the chain rule to the exponential function. The label should be 'chain' to accurately reflect the rule applied. Additionally, Step 10's note is confusing and self-contradictory, though the algebraic transition is valid.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two rules at once: it differentiates the exponential using the chain rule and simultaneously applies the exponential rule. This violates the rule that each step must change only one thing. The correct labeling would require a separate step for the chain rule or a single step labeled "exponential" that includes the inner derivative, but as written it combines two operations.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, chain rule, and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies the chain rule (derivative of exp(x‑3) = exp(x‑3)·(x‑3)′) but is labeled "exponential". The correct label is "chain". This mislabeling violates the rule‑name requirement.
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.