Derivative of \( \displaystyle \left(x - 3\right)^{\sqrt{x - 3}} \)
Problem 2.825 · hard
Differentiate \( \displaystyle f(x) = \left(x - 3\right)^{\sqrt{x - 3}} \).
- \[ \frac{d}{d x} \left(x - 3\right)^{\sqrt{x - 3}} \]rewriteRewrite the base and exponent using the exponential and logarithm functions.✓ Proved
- \[ = \frac{d}{d x} e^{\sqrt{x - 3} \ln{\left(x - 3 \right)}} \]derivativeApply the derivative rule for the exponential function.≈ Checked numerically
- \[ = e^{\sqrt{x - 3} \ln{\left(x - 3 \right)}} \frac{d}{d x} \sqrt{x - 3} \ln{\left(x - 3 \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\sqrt{x - 3} \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \sqrt{x - 3}\right) e^{\sqrt{x - 3} \ln{\left(x - 3 \right)}} \]productApply the product rule to the terms in the parentheses.✓ Proved
- \[ = \left(\frac{\ln{\left(x - 3 \right)}}{2 \sqrt{x - 3}} + \frac{1}{\sqrt{x - 3}}\right) e^{\sqrt{x - 3} \ln{\left(x - 3 \right)}} \]chain algebraDifferentiate the square root and logarithmic functions using the chain rule. Simplify the terms inside the parentheses.✓ Proved
- \[ = \left(x - 3\right)^{\sqrt{x - 3}} \left(\frac{\ln{\left(x - 3 \right)}}{2 \sqrt{x - 3}} + \frac{1}{\sqrt{x - 3}}\right) \]simplifyConvert the exponential form back to the original power form.≈ Checked numerically
Answer \( \frac{\left(x - 3\right)^{\sqrt{x - 3} - \frac{1}{2}} \left(\ln{\left(x - 3 \right)} + 2\right)}{2} \)
Lines: 6 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left ((x - 3)**(sqrt(x - 3)) - exp(sqrt(x - 3)*log(x - 3)))*(log(x - 3) + 2)/(2*sqrt(x - 3)); numeric agreement only, at 24 of 24 sampled points 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 undefined where x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (-(x - 3)**(sqrt(x - 3)) + exp(sqrt(x - 3)*log(x - 3)))*(log(x - 3) + 2)/(2*sqrt(x - 3)); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x - 3 = 0 |
| 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: fail (style) — Step 3 incorrectly labels the application of the chain rule for the exponential function as a product rule. The derivative of exp(u) is exp(u)·u', which is a chain rule application, not a product rule. The label should be "chain" instead of "product".qwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'derivative' but performs an algebraic rewrite of the base/exponent form into exponential/logarithmic form; the actual differentiation happens in Step 3. Step 5 is labeled 'chain' but applies the derivative rules for sqrt and log (which are 'derivative' or 'logarithmic'/'power' steps) rather than just unwrapping a chain structure, and it combines multiple differentiation actions into one step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 2 is labeled 'derivative' but performs an algebraic rewrite of the base/exponent form into exponential/logarithmic form; the actual differentiation happens in Step 3. Step 5 is labeled 'chain' but applies the derivative rules for sqrt and log (which are 'derivative' or 'logarithmic'/'power' steps) rather than just unwrapping a chain structure, and it combines multiple differentiation actions into one step.gpt-oss:20b: fail (style) 2026-09-26 — Step 3 incorrectly labels the application of the chain rule for the exponential function as a product rule. The derivative of exp(u) is exp(u)·u', which is a chain rule application, not a product rule. The label should be "chain" instead of "product".qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 2 is labeled 'derivative' but performs no differentiation; it merely rewrites the expression, which should be labeled 'rewrite'. Step 5 is labeled 'chain' but applies the derivative rules for sqrt and log (which are 'derivative' or 'logarithmic'/'power' depending on granularity, but certainly not just 'chain' which implies composition handling, and it combines two distinct derivative calculations into one step).gpt-oss:20b: fail (error) 2026-09-26 — Step 3 incorrectly labels the application of the chain rule for the exponential function as a product rule. The derivative of exp(u) is exp(u)*u', which is a chain rule application, not a product rule. This mislabeling constitutes an error in the solution.
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.