Derivative of \( \displaystyle \left(9 x^{2} + 1\right)^{3 x} \)
Problem 2.408 · hard Video
Differentiate \( \displaystyle f(x) = \left(9 x^{2} + 1\right)^{3 x} \).
- \[ \frac{d}{d x} \left(9 x^{2} + 1\right)^{3 x} \]rewriteRewrite the function using the exponential and logarithm identity.✓ Proved
- \[ = \frac{d}{d x} e^{3 x \ln{\left(9 x^{2} + 1 \right)}} \]derivativeApply the derivative to the exponential function.✓ Proved
- \[ = e^{3 x \ln{\left(9 x^{2} + 1 \right)}} \frac{d}{d x} 3 x \ln{\left(9 x^{2} + 1 \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(3 x \frac{d}{d x} \ln{\left(9 x^{2} + 1 \right)} + \ln{\left(9 x^{2} + 1 \right)} \frac{d}{d x} 3 x\right) e^{3 x \ln{\left(9 x^{2} + 1 \right)}} \]productApply the product rule to the two terms in the parentheses.✓ Proved
- \[ = \left(3 x \frac{d}{d x} \ln{\left(9 x^{2} + 1 \right)} + 3 \ln{\left(9 x^{2} + 1 \right)}\right) e^{3 x \ln{\left(9 x^{2} + 1 \right)}} \]constant-multipleDifferentiate 3*x.✓ Proved
- \[ = \left(\frac{3 x \frac{d}{d x} \left(9 x^{2} + 1\right)}{9 x^{2} + 1} + 3 \ln{\left(9 x^{2} + 1 \right)}\right) e^{3 x \ln{\left(9 x^{2} + 1 \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{54 x^{2}}{9 x^{2} + 1} + 3 \ln{\left(9 x^{2} + 1 \right)}\right) e^{3 x \ln{\left(9 x^{2} + 1 \right)}} \]derivative algebraDifferentiate the inner function of the logarithm. Simplify the expression inside the parentheses.✓ Proved
- \[ = \left(9 x^{2} + 1\right)^{3 x} \left(\frac{54 x^{2}}{9 x^{2} + 1} + 3 \ln{\left(9 x^{2} + 1 \right)}\right) \]simplifySubstitute the original function back for the exponential term.✓ Proved
Answer \( \left(9 x^{2} + 1\right)^{3 x} \left(\frac{54 x^{2}}{9 x^{2} + 1} + 3 \log{\left(9 x^{2} + 1 \right)}\right) \)
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. 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 9*x**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 9*x**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 9*x**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 9*x**2 + 1 = 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 undefined where 9*x**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: fail (style) — Step 3 incorrectly labels the application of the chain rule for the exponential as a product rule. The derivative of exp(u) is exp(u)·u', which is a chain rule application, not a product rule. The subsequent steps correctly apply the product rule to 3*x·log(...).qwen3.6:27b-mlx: pass — The solution correctly applies logarithmic differentiation. Each step changes only one aspect of the expression, and the labels accurately reflect the rules applied (e.g., 'chain' for the log derivative, 'product' for the exponent expansion).
Senior review claude-sonnet-5-5, 2026-10-08: fail — Every line is mathematically correct and SymPy proved each step and the final answer. Line 3 is mislabeled 'product' with the note 'apply the product rule to the exponent', when the step is the exponential/chain rule. Line 2 is labeled 'derivative' although the step from line 1 is a rewrite, so the labels at the start are off.
gpt-oss:20b: uphold — Going from line 2 to line 3 turns Derivative(exp(u)) into exp(u)*Derivative(u), which is the exponential/chain rule. The product rule is only applied in the next step (line 3 to line 4), so the label 'product' names a rule this step did not use.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-08 — Every line is mathematically correct and SymPy proved each step and the final answer. Line 3 is mislabeled 'product' with the note 'apply the product rule to the exponent', when the step is the exponential/chain rule. Line 2 is labeled 'derivative' although the step from line 1 is a rewrite, so the labels at the start are off.qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies logarithmic differentiation. Each step changes only one aspect of the expression, and the labels accurately reflect the rules applied (e.g., 'chain' for the log derivative, 'product' for the exponent expansion).gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule for the exponential as a product rule. The derivative of exp(u) is exp(u)·u', which is a chain rule application, not a product rule. The subsequent steps correctly apply the product rule to 3*x·log(...).qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 is labeled 'derivative' but performs no differentiation; it merely sets up the expression for the next step, so 'rewrite' is the correct label. Step 5 is labeled 'constant-multiple' but performs the differentiation of 3x, so 'derivative' is the correct label.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule for the exponential as a product rule. The derivative of exp(u) should be identified as a chain rule step, not a product rule.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule for the derivative of exp(3*x*log(9*x**2 + 1)) as a product rule. The correct rule name is "chain", not "product". This mislabeling misleads a student about which rule is being applied.
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.