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Leibniz's rule for the \( n \)th derivative of a product \( uv \) reads

  • A.\( u^{(n)} v + u v^{(n)} \) with no further terms
  • B.\( n\,u^{(n-1)} v^{(n-1)} \) with no further terms
  • C.\( \sum_{r=0}^{n} \binom{n}{r} u^{(n-r)} v^{(r)} \)
  • D.\( u^{(n)} v^{(n)} \) with no further terms

Answer: C

Repeated use of the product rule generates binomial coefficients, exactly as in the binomial theorem:

\[ (uv)^{(n)} \;=\; \sum_{r=0}^{n} \binom{n}{r} u^{(n-r)} v^{(r)}. \]

For \( n = 1 \) this is \( u'v + uv' \). The two-term forms offered omit every mixed derivative between the extremes, which for \( n = 2 \) loses the \( 2u'v' \) term

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