Derivative of \( \displaystyle \left(3 x + 2\right)^{3 x + 1} \)
Problem 2.386 · medium
Differentiate \( \displaystyle f(x) = \left(3 x + 2\right)^{3 x + 1} \).
- \[ \frac{d}{d x} \left(3 x + 2\right)^{3 x + 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \]rewriteRewrite the function using the exponential and logarithm form.≈ Checked numerically
- \[ = e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \frac{d}{d x} \left(3 x + 1\right) \ln{\left(3 x + 2 \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\left(3 x + 1\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 1\right)\right) e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \]productApply the product rule to the exponent's derivative.✓ Proved
- \[ = \left(\left(3 x + 1\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 3 \ln{\left(3 x + 2 \right)}\right) e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(\frac{\left(3 x + 1\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} + 3 \ln{\left(3 x + 2 \right)}\right) e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{3 \left(3 x + 1\right)}{3 x + 2} + 3 \ln{\left(3 x + 2 \right)}\right) e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \]derivativeDifferentiate the inner function of the logarithm.✓ Proved
- \[ = \left(3 \ln{\left(3 x + 2 \right)} + \frac{9 x + 3}{3 x + 2}\right) e^{\left(3 x + 1\right) \ln{\left(3 x + 2 \right)}} \]algebraSimplify the expression algebraically.✓ Proved
- \[ = \left(3 x + 2\right)^{3 x + 1} \left(3 \ln{\left(3 x + 2 \right)} + \frac{9 x + 3}{3 x + 2}\right) \]simplifyConvert the exponential form back to the original power form.≈ Checked numerically
Answer \( 3 \left(3 x + 2\right)^{3 x} \left(3 x + \left(3 x + 2\right) \log{\left(3 x + 2 \right)} + 1\right) \)
✓ Nihil obstat Lines: 8 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*((3*x + 2)**(3*x + 1) - exp((3*x + 1)*log(3*x + 2)))*(3*x + (3*x + 2)*log(3*x + 2) + 1)/(3*x + 2); 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*(-(3*x + 2)**(3*x + 1) + exp((3*x + 1)*log(3*x + 2)))*(3*x + (3*x + 2)*log(3*x + 2) + 1)/(3*x + 2); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 3*x + 2 = 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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, product rule, and logarithmic differentiation steps. Each step isolates a single rule application, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies logarithmic differentiation, breaking down the chain and product rules into single-rule steps as required. The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-09-20
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.