Home›Calculus 1›Logarithmic differentiation›Problem 2.825

Derivative of \( \displaystyle \left(x - 3\right)^{\sqrt{x - 3}} \)

Problem 2.825 · hard Video

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 independently
\[ = e^{\sqrt{x - 3} \ln{\left(x - 3 \right)}} \frac{d}{d x} \sqrt{x - 3} \ln{\left(x - 3 \right)} \]
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 independently

Answer \( \frac{\left(x - 3\right)^{\sqrt{x - 3} - \frac{1}{2}} \left(\ln{\left(x - 3 \right)} + 2\right)}{2} \)

Top: the graph of f = (x - 3)^(sqrt(x - 3)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (x - 3)^(sqrt(x - 3))·(log(x - 3)/(2sqrt(x - 3)) + 1/sqrt(x - 3)).

Lines: 6 proved, 2 checked by independent solution. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, 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)/(2sqrt(x - 3)); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: Pow b*s rewritten as exp(slog b) (definition of principal power) gives line 2 integrand exactly; derivative of equal expressions is equal (`checks/2.825/line2-81e542fc.py`) 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 independently	claude-sonnet-5-5 script, run by sympy 1.14.0, 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)/(2sqrt(x - 3)); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: replacing (x-3)*sqrt(x-3) by exp(sqrt(x-3)log(x-3)) in line 7 gives line 6 (difference simplifies to 0) (`checks/2.825/line7-62f690a5.py`) 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 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 could mislead a student about which rule to use.
qwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'derivative' but performs a rewrite of the function form (using exp and log) rather than applying a differentiation rule. Step 5 is labeled 'chain' but applies the chain rule to two separate terms simultaneously, violating the one-rule-per-step constraint.

Senior review claude-sonnet-5-5, 2026-10-13: fail — The calculus and the final answer are correct, and every line is a valid equivalence. The rule labels on lines 2 and 3 are shifted: the rewrite to exp/log is labelled 'derivative' and the exp derivative is labelled 'product', which would teach the wrong rule name.

gpt-oss:20b: uphold — The step from line 2 to line 3 is the derivative of exp(u), i.e. exp(u)*u' (exponential/chain), yet line 3 is labelled 'product'; the product rule only appears in the next step.
qwen3.6:27b-mlx: uphold — The step into line 2 only rewrites (x-3)**sqrt(x-3) as exp(sqrt(x-3)*log(x-3)), so 'derivative' is the wrong label (it should be 'rewrite'); the complaint about step 5 is wrong, since applying the chain rule to two terms in one step counts as one change.

Every verdict on record (7)

claude-sonnet-5-5: fail (misleading) 2026-10-13 — The calculus and the final answer are correct, and every line is a valid equivalence. The rule labels on lines 2 and 3 are shifted: the rewrite to exp/log is labelled 'derivative' and the exp derivative is labelled 'product', which would teach the wrong rule name.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 2 is labeled 'derivative' but performs a rewrite of the function form (using exp and log) rather than applying a differentiation rule. Step 5 is labeled 'chain' but applies the chain rule to two separate terms simultaneously, violating the one-rule-per-step constraint.
gpt-oss:20b: fail (style) 2026-09-27 — Step 3 incorrectly labels the application of the chain rule 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 could mislead a student about which rule to use.
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 2026-09-27 with SymPy 1.14.0.