Home›Calculus 1›Derivatives of exponential functions›Problem 2.577

Derivative of \( \displaystyle \sqrt{5} \sqrt{x} e^{5 x} \)

Problem 2.577 · hard Video

Differentiate \( \displaystyle f(x) = \sqrt{5} \sqrt{x} e^{5 x} \).

\[ \frac{d}{d x} \sqrt{5} \sqrt{x} e^{5 x} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \sqrt{5} \frac{d}{d x} \sqrt{x} e^{5 x} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \sqrt{5} \left(\sqrt{x} \frac{d}{d x} e^{5 x} + e^{5 x} \frac{d}{d x} \sqrt{x}\right) \]
productApply the product rule.✓ Proved
\[ = \sqrt{5} \left(\sqrt{x} \frac{d}{d x} e^{5 x} + e^{5 x} \frac{d}{d x} x^{0.5}\right) \]
rewriteRewrite sqrt(x) as x**0.5.✓ Proved
\[ = \sqrt{5} \left(\frac{0.5 e^{5 x}}{x^{0.5}} + \sqrt{x} \frac{d}{d x} e^{5 x}\right) \]
derivativeDifferentiate x**0.5.✓ Proved
\[ = \sqrt{5} \left(\frac{0.5 e^{5 x}}{x^{0.5}} + 5 \sqrt{x} e^{5 x}\right) \]
derivativeDifferentiate exp(5*x) using the chain rule.✓ Proved
\[ = \sqrt{5} \left(\frac{0.5}{x^{0.5}} + 5 \sqrt{x}\right) e^{5 x} \]
algebraFactor out exp(5*x).✓ Proved
\[ = \sqrt{5} \left(5 \sqrt{x} + \frac{1}{2 \sqrt{x}}\right) e^{5 x} \]
algebraRewrite 0.5*x**(-0.5) as 1/(2*sqrt(x)).✓ Proved
\[ = \frac{\sqrt{5} \left(10 x + 1\right) e^{5 x}}{2 \sqrt{x}} \]
simplifyCombine the terms into a single fraction.✓ Proved

Answer \( \frac{\sqrt{5} \left(10 x + 1\right) e^{5 x}}{2 \sqrt{x}} \)

Top: the graph of f = sqrt(5)·sqrt(x)·e^(5x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5sqrt(5)·sqrt(x)·e^(5x) + sqrt(5)·e^(5x)/(2sqrt(x)).

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — Step 6 is labeled 'derivative' but applies the chain rule to differentiate exp(5x); the label should be 'chain'. Step 4 rewrites sqrt(x) as x**0.5, which is a valid rewrite, but the subsequent differentiation in Step 5 is labeled 'derivative' which is acceptable for power rule application, though 'power' would be more precise if available; however, the primary defect is the mislabeling of the chain rule application in Step 6.

Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is an equivalent expression, each step makes one change, and the labels are acceptable. The final line matches the stated answer. The 0.5 floats in steps 4-8 are a cosmetic quirk, not a defect.

qwen3.6:27b-mlx: dismiss — Step 6 evaluates Derivative(exp(5*x)) to 5*exp(5*x), and 'derivative' is a legitimate label for that evaluation. The contract also treats evaluating an inner derivative as part of the chain-rule step, so neither label is wrong. The comment on Step 5 ('power' would be more precise) is only a preference, and 'derivative' is acceptable there too.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — Every line is an equivalent expression, each step makes one change, and the labels are acceptable. The final line matches the stated answer. The 0.5 floats in steps 4-8 are a cosmetic quirk, not a defect.
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 6 is labeled 'derivative' but applies the chain rule to differentiate exp(5x); the label should be 'chain'. Step 4 rewrites sqrt(x) as x**0.5, which is a valid rewrite, but the subsequent differentiation in Step 5 is labeled 'derivative' which is acceptable for power rule application, though 'power' would be more precise if available; however, the primary defect is the mislabeling of the chain rule application in Step 6.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 6 is labeled 'derivative' but applies the chain rule to differentiate exp(5*x); the label should be 'chain'. Additionally, the note explicitly mentions the chain rule while the label does not, creating a contradiction that would confuse a student about which label corresponds to which operation.
gpt-oss:20b: pass 2026-09-21

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.