Derivative of \( \displaystyle \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \)
Problem 2.324 · hard
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \).
- \[ \frac{d}{d x} \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \]rewriteRewrite the base-exponent function using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]chainApply the chain rule.≈ Checked numerically
- \[ = e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \frac{d}{d x} \left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)} \]productApply the product rule.✓ Proved
- \[ = \left(\left(5 x + 2\right) \frac{d}{d x} \ln{\left(1 + \frac{1}{5 x + 2} \right)} + \ln{\left(1 + \frac{1}{5 x + 2} \right)} \frac{d}{d x} \left(5 x + 2\right)\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]derivativeDifferentiate both parts of the product.✓ Proved
- \[ = \left(\left(5 x + 2\right) \frac{d}{d x} \ln{\left(1 + \frac{1}{5 x + 2} \right)} + 5 \ln{\left(1 + \frac{1}{5 x + 2} \right)}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]derivativeDifferentiate the first factor.✓ Proved
- \[ = \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} + \frac{\left(5 x + 2\right) \frac{d}{d x} \left(1 + \frac{1}{5 x + 2}\right)}{1 + \frac{1}{5 x + 2}}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} + \frac{\left(5 x + 2\right) \left(\frac{d}{d x} 1 + \frac{d}{d x} \frac{1}{5 x + 2}\right)}{1 + \frac{1}{5 x + 2}}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} + \frac{\left(5 x + 2\right) \frac{d}{d x} \frac{1}{5 x + 2}}{1 + \frac{1}{5 x + 2}}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]constantThe derivative of the constant 1 is 0.✓ Proved
- \[ = \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} - \frac{\frac{d}{d x} \left(5 x + 2\right)}{\left(1 + \frac{1}{5 x + 2}\right) \left(5 x + 2\right)}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]chainDifferentiate the reciprocal function.✓ Proved
- \[ = \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} - \frac{5}{\left(1 + \frac{1}{5 x + 2}\right) \left(5 x + 2\right)}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]derivative algebra algebraDifferentiate the linear term 5*x + 2. Simplify the expression. Cancel (5*x + 2) in the numerator and denominator.✓ Proved
- \[ = \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} - \frac{5}{5 x + 3}\right) e^{\left(5 x + 2\right) \ln{\left(1 + \frac{1}{5 x + 2} \right)}} \]algebra algebraSimplify the complex fraction. Cancel (5*x + 2) in the numerator and denominator.✓ Proved
- \[ = \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \left(5 \ln{\left(1 + \frac{1}{5 x + 2} \right)} - \frac{5}{5 x + 3}\right) \]rewriteConvert the exponential form back to the original power form.≈ Checked numerically
- \[ = \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \left(5 \ln{\left(\frac{5 x + 3}{5 x + 2} \right)} - \frac{5}{5 x + 3}\right) \]simplifySimplify the logarithm.✓ Proved
Answer \( \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \left(5 \log{\left(1 + \frac{1}{5 x + 2} \right)} - \frac{5}{\left(1 + \frac{1}{5 x + 2}\right) \left(5 x + 2\right)}\right) \)
Lines: 15 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 5*(((5*x + 3)/(5*x + 2))**(5*x + 2)*((5*x + 3)*log((5*x + 3)/(5*x + 2)) - 1) + (-(5*x + 3)*log((5*x + 3)/(5*x + 2)) + 1)*exp((5*x + 2)*log((5*x + 3)/(5*x + 2))))/(5*x + 3); numeric agreement only, at 24 of 24 sampled points undefined where 5*x + 2 = 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 undefined where 5*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 0 undefined where 5*x + 3 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 5*x + 3 = 0 |
| 15 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 5*(((5*x + 3)/(5*x + 2))**(5*x + 2)*(-(5*x + 3)*log((5*x + 3)/(5*x + 2)) + 1) + ((5*x + 3)*log((5*x + 3)/(5*x + 2)) - 1)*exp((5*x + 2)*log((5*x + 3)/(5*x + 2))))/(5*x + 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 5*x + 2 = 0 undefined where 5*x + 3 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 3 = 0 undefined where 5*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 undefined where 5*x + 2 = 0 undefined where 1 + 1/(5*x + 2) = 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) — Several steps apply more than one rule at once or mislabel the rule used. For example, step 3 applies the chain rule (derivative of exp) but is labeled “product”; step 4 incorrectly applies the product rule to a single function; and many later steps combine algebraic simplifications with derivative steps without separate labels. These labeling errors violate the contract’s requirement that each step change only one thing and use the correct rule name.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'chain' but performs no differentiation; it merely rewrites the expression from step 1 into exponential form, which should be labeled 'rewrite'. Step 4 is labeled 'derivative' but applies the product rule to split the derivative of a product, which should be labeled 'product'. Step 9 is labeled 'chain' but applies the quotient rule (or power rule with negative exponent) to differentiate a reciprocal, which is not the chain rule.
Every verdict on record (13)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 is labeled 'chain' but performs no differentiation; it merely rewrites the expression from step 1 into exponential form, which should be labeled 'rewrite'. Step 4 is labeled 'derivative' but applies the product rule to split the derivative of a product, which should be labeled 'product'. Step 9 is labeled 'chain' but applies the quotient rule (or power rule with negative exponent) to differentiate a reciprocal, which is not the chain rule.gpt-oss:20b: fail (style) 2026-09-20 — Several steps apply more than one rule at once or mislabel the rule used. For example, step 3 applies the chain rule (derivative of exp) but is labeled “product”; step 4 incorrectly applies the product rule to a single function; and many later steps combine algebraic simplifications with derivative steps without separate labels. These labeling errors violate the contract’s requirement that each step change only one thing and use the correct rule name.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 4 violates the 'one rule per step' constraint by applying the product rule to the outer structure and simultaneously differentiating the first factor (5*x + 2) into 5. These are two distinct operations that should be separated.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels a chain‑rule application as a product rule, and step 4 similarly mislabels the differentiation of the inner product. The algebraic manipulations themselves are correct, but the rule names are inaccurate.qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 4 applies the product rule to the entire expression, but the label 'derivative' is incorrect; it should be 'product'. Step 5 differentiates only one factor of the product while leaving the other term's derivative unevaluated, which is a partial application not covered by a single standard rule label like 'derivative' or 'constant-multiple' in isolation without context, but more critically, Step 4's label is definitively wrong for the operation performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 3 incorrectly labels the chain rule as a product rule, and step 4 (and step 5) incorrectly label the product rule as a derivative step. Each of these steps applies more than one rule or mislabels the rule used, violating the contract.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the exponential rewrite, chain rule, and product rule in distinct steps. The algebraic simplifications are valid and clearly labeled.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 3 incorrectly applies the product rule to the derivative of exp((5*x+2)*log(1+1/(5*x+2)))—the correct rule is the chain rule for exp, not a product rule. This single step applies two rules at once and mislabels the operation.qwen3.6:27b-mlx: fail (style) 2026-09-18 — Step 4 applies the product rule to the entire inner expression, but the label 'derivative' is incorrect; it should be 'product'. Step 5 then differentiates only one part of the product, which is a valid sub-step, but the previous step's labeling was wrong. More critically, Step 4 claims to differentiate 'both parts' but only sets up the product rule structure without actually performing the differentiation of the factors in that single step, making the label 'derivative' (which implies unfolding d/dx on a known form) misleading compared to 'product'. However, the most glaring issue is Step 4's note says 'Differentiate both parts' but the output is just the product rule expansion, not the result of differentiation. The label 'derivative' is not in the list for product rule expansion. The label should be 'product'. Step 5 is labeled 'derivative' but it differentiates only the first term, leaving the second term as a derivative operator. This is a valid step if labeled 'derivative' (unfolding d/dx(5x+2)), but Step 4 is definitely mislabeled. Let's look closer. Step 4: `Derivative(u*v) -> u'v + uv'`. This is the product rule. The label is 'derivative'. This is a defect. Step 5: `u'v + uv' -> 5*v + u*Derivative(log...)`. This differentiates the first term. Label 'derivative' is acceptable for unfolding d/dx(5x+2)=5. Step 6: Differentiates the log term using chain rule. Label 'chain' is correct. Step 7: Differentiates the sum inside. Label 'sum' is correct. Step 8: Derivative of constant 1. Label 'constant' is correct. Step 9: Derivative of reciprocal. Label 'chain' is acceptable (or quotient/algebra). Step 10: Derivative of linear term. Label 'derivative' is correct. Step 11-14: Algebraic simplification. Labels 'algebra' are correct. Step 15: Rewrite exp back to power. Label 'rewrite' is correct. Step 16: Simplify log. Label 'simplify' is correct. The main defect is Step 4 labeled 'derivative' instead of 'product'.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (error) 2026-09-18 — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(g(x)) is exp(g(x))*g'(x), not a product of two separate factors. This mislabeling violates the rule‑granularity 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.