𝑑
𝑑
𝑥
[
𝑒
𝑥
𝑥
2
+
4
]
=

Question

BlackTom AI · Accepted Answer

The question presents a derivative task: compute d/dx of e^x / (x^2 + 4). To approach this, recall the quotient rule: if y = u/v, then y' = (u'v − uv') / v^2.

Option 1: 'e^x 2x' — This appears to be missing the correct structure of the derivative. It does not align with the quotient rule form and lacks the complete numerator pattern that combines u'v and uv'. It is insufficient because it neither shows the subtraction of two large terms nor a denominator squared, which is essential in the quotient rule.

Option 2: '2x e^x − e^x(x^2 + 4)(x^2 + 4)2' — This expression seems to mix factors in a way that doesn't reflect the clean quotient rule structure. It appears to misplace terms and includes an extra (x^2+4) factor in the second term that would complicate the denominator in an incorrect manner. The overall sign and arrangement do not match the standard form (u'v − uv') / v^2.

Option 3: '2x e^x' — This is far too simplistic. It resembles a derivative of e^x times something linear in x, but it ignores the division by (x^2+4) and the need to apply the quotient rule correctly. It fails to account for the impact of both the numerator's derivative and the denominator.

Option 4: 'e^x(x^2+4) − 2x e^x(x^2+4)2' — This option hints at using the quotient rule by presenting a subtraction between two terms that involve e^x and (x^2+4), but the exact formatting is ambiguous. The intended correct structure would be something like [e^x(x^2+4) − e^x(2x)] / (x^2+4)^2, which simplifies to e^x[(x^2+4) − 2x] / (x^2+4)^2. The presence of a squared denominator is essential. If this option is meant to reflect that standard form, it would be the correct scaffold, albeit the text here isn't perfectly clear about the denominator.

In summary:
- The derivative of e^x / (x^2 + 4) requires the quotient rule, giving [e^x(x^2+4) − e^x(2x)] / (x^2+4)^2 = e^x[(x^2 − 2x + 4)] / (x^2+4)^2. 
- The incorrect options fail to match this required structure either due to missing the subtraction term, incorrect factors, or missing the squared denominator. 
- The correct choice, if interpreted as incorporating the proper numerator structure e^x(x^2+4) − 2x e^x, over (x^2+4)^2, aligns with the quotient rule result and the simplification to e^x[(x^2 − 2x + 4)]/(x^2+4)^2.

MATH_1225_17255_202501 3.2 The Product and Quotient Rules (timed)

𝑑 𝑑 𝑥 [ 𝑒 𝑥 𝑥 2 + 4 ] =

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