Derivative of \( \displaystyle \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \)
Problem 2.380 · hard
Differentiate \( \displaystyle f(x) = \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \).
- \[ \frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]rewriteRewrite the function using the exponential and logarithm form.✓ Proved
- \[ = e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \frac{d}{d x} \left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\left(5 x + 1\right) \frac{d}{d x} \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} + \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\left(5 x + 1\right) \frac{d}{d x} \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(\frac{\left(5 x + 1\right) \frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{\left(5 x + 1\right) \left(50 x + 10\right)}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]derivative derivativeDifferentiate the inner function of the logarithm. Differentiate the inner function of the power.✓ Proved
- \[ = \left(\frac{10 \left(5 x + 1\right)^{2}}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) e^{\left(5 x + 1\right) \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}} \]algebraSimplify the expression inside the parentheses.✓ Proved
- \[ = \left(\frac{10 \left(5 x + 1\right)^{2}}{\left(5 x + 1\right)^{2} + 1} + 5 \ln{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \]simplifyConvert the exponential form back to the original base.✓ Proved
Answer \( 5 \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x} \left(2 \left(5 x + 1\right)^{2} + \left(\left(5 x + 1\right)^{2} + 1\right) \log{\left(\left(5 x + 1\right)^{2} + 1 \right)}\right) \)
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. 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 | ✓ 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 (5*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where (5*x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where (5*x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where (5*x + 1)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where (5*x + 1)**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 |
| 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: fail (error) — Step 7 claims to differentiate the inner function of the logarithm, but the expression changes from `Derivative((5*x + 1)**2 + 1, x)` to `2*(5*x + 1)*5`. This is a differentiation step, but the label is 'derivative' which is acceptable for unfolding d/dx. However, Step 8 changes `2*(5*x + 1)*5` to `10*(5*x + 1)`. This is an algebraic simplification (multiplication), not a differentiation. The label 'derivative' is incorrect for Step 8. Furthermore, Step 7 applies the power rule `d/dx(u^2) = 2u*u'` and the constant multiple rule `d/dx(5x+1)=5` simultaneously. While 'derivative' covers the differentiation, the step performs two distinct differentiation operations (power rule on the square, and linear derivative on the inner term) in one go, which violates the 'one thing' rule if interpreted strictly, but more critically, Step 8 is definitely not a derivative step.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 7 claims to differentiate the inner function of the logarithm, but the expression changes from `Derivative((5*x + 1)**2 + 1, x)` to `2*(5*x + 1)*5`. This is a differentiation step, but the label is 'derivative' which is acceptable for unfolding d/dx. However, Step 8 changes `2*(5*x + 1)*5` to `10*(5*x + 1)`. This is an algebraic simplification (multiplication), not a differentiation. The label 'derivative' is incorrect for Step 8. Furthermore, Step 7 applies the power rule `d/dx(u^2) = 2u*u'` and the constant multiple rule `d/dx(5x+1)=5` simultaneously. While 'derivative' covers the differentiation, the step performs two distinct differentiation operations (power rule on the square, and linear derivative on the inner term) in one go, which violates the 'one thing' rule if interpreted strictly, but more critically, Step 8 is definitely not a derivative step.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 7 claims to differentiate the inner function of the logarithm, but the expression `(5*x + 1)**2 + 1` is not yet differentiated; the derivative `2*(5*x + 1)*5` appears in Step 7 without the intermediate step of differentiating the square. Furthermore, Step 8 is redundant and mislabeled as 'derivative' when it merely simplifies the constant multiplication `2*5` to `10`, which should be 'algebra' or 'simplify'. The primary defect is that Step 7 performs the differentiation of `(5*x+1)^2` implicitly within the step labeled for the outer chain rule application, violating the one-rule-per-step constraint by combining the chain rule application for the log with the power rule differentiation inside.gpt-oss:20b: fail (style) 2026-09-20 — Step 8 applies the label "derivative" but only performs a simplification of the expression 2*(5*x+1)*5 to 10*(5*x+1). This violates the rule that each step must apply exactly one rule and label it correctly.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 7 claims to differentiate the inner function of the logarithm, but the term (5*x + 1)**2 is not yet differentiated; it remains as (5*x + 1)**2. Step 8 then differentiates (5*x + 1)**2 to 10*(5*x + 1), but labels this step as 'derivative' while also performing algebraic simplification of the constant factor 2*5=10, or rather, it applies the power rule and chain rule in one go without proper decomposition. More critically, Step 7's note says 'Differentiate the inner function of the logarithm' but the expression in Step 7 still contains Derivative((5*x + 1)**2 + 1, x) which is not evaluated. Wait, looking closely at Step 7: it replaces Derivative(log(...), x) with (1/...)*Derivative((5*x+1)**2+1, x). This is the chain rule for log, labeled 'derivative'. That is acceptable. Step 8 replaces Derivative((5*x+1)**2+1, x) with 10*(5*x+1). This requires differentiating (5*x+1)**2. The derivative of (5*x+1)**2 is 2*(5*x+1)*5 = 10*(5*x+1). Step 8 labels this 'derivative'. This is a single differentiation step. However, Step 7 and 8 are split oddly. Step 7 applies chain rule to log. Step 8 applies chain rule/power rule to the inner quadratic. The label 'derivative' is used for both. This is acceptable. Let's look closer at Step 7. The transition from 6 to 7 is: Derivative(log(u), x) -> (1/u)*Derivative(u, x). This is the chain rule. The label is 'derivative'. This is a valid label for 'unfolding d/dx on a known form' if we consider d/dx log(u) as a known form, but usually 'chain' is preferred for composite functions. The contract says 'derivative' means unfolding d/dx on a known form (sin, exp, etc.). Log is in the list. So 'derivative' is acceptable for d/dx log(u). Step 8: Derivative((5*x+1)**2+1, x) -> 10*(5*x+1). This is d/dx (u^2+1) = 2u*u'. This is chain rule + power rule. The label is 'derivative'. Is this 'unfolding d/dx on a known form'? The form is (linear)^2. It's a composite. Usually 'chain' is needed. But 'derivative' might cover it if it's considered a standard derivative. Let's look for a clearer error. Step 9: Simplifies (5*x+1)*(10*(5*x+1)) to 10*(5*x+1)^2. This is algebra. Label 'algebra'. Correct. Step 10: Converts exp(...) back. Label 'simplify'. Correct. Re-evaluating Step 7 and 8. Step 6 has: ... + (5*x + 1)*(1/((5*x + 1)**2 + 1))*Derivative((5*x + 1)**2 + 1, x) Step 7 has: ... + (5*x + 1)*(1/((5*x + 1)**2 + 1))*(2*(5*x + 1)*5) Wait, Step 7's expression in the prompt is: `exp(...) * (5*log(...) + (5*x + 1)*(1/((5*x + 1)**2 + 1))*(2*(5*x + 1)*5))` Step 8's expression is: `exp(...) * (5*log(...) + (5*x + 1)*(1/((5*x + 1)**2 + 1))*(10*(5*x + 1)))` Step 7 calculates the derivative of (5*x+1)**2 as 2*(5*x+1)*5. This is applying the chain rule/power rule. The label is 'derivative'. Step 8 simplifies 2*5 to 10. The label is 'derivative'. This is WRONG. Step 8 is purely algebraic simplification (2*5=10). It does not perform any differentiation. The derivative was already computed in Step 7. Step 8 just simplifies the constant coefficient. Labeling an algebraic simplification as 'derivative' is a defect. Also, Step 7 applies the chain rule to (5*x+1)**2. The label 'derivative' is acceptable for 'unfolding d/dx on a known form' if we consider the derivative of a composite polynomial as a standard form, but typically 'chain' is better. However, the error in Step 8 is clear: it performs no differentiation, only arithmetic simplification, yet is labeled 'derivative'.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.