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The functions p(x) p(x) and q(x) q(x) are differentiable. The function v(x) v(x) is defined as: v(x)=p(x)q(x) v(x)= \frac{p(x)}{q(x)} If p(8)=4 p(8)= 4 , p′(8)=2 p'(8)= 2 , q(8)=3 q(8)= 3 , and q′(8)=−1 q'(8)= -1 , what is v′(8) v'(8) ? Simplify any fractions. v′(8)= v'(8)=

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Q. The functions p(x) p(x) and q(x) q(x) are differentiable. The function v(x) v(x) is defined as: v(x)=p(x)q(x) v(x)= \frac{p(x)}{q(x)} If p(8)=4 p(8)= 4 , p′(8)=2 p'(8)= 2 , q(8)=3 q(8)= 3 , and q′(8)=−1 q'(8)= -1 , what is v′(8) v'(8) ? Simplify any fractions. v′(8)= v'(8)=
  1. Identify the rule: Identify the rule for differentiating a quotient.\newlineThe quotient rule for differentiation states that if v(x)=p(x)q(x)v(x) = \frac{p(x)}{q(x)}, then v′(x)=p′(x)q(x)−p(x)q′(x)(q(x))2v'(x) = \frac{p'(x)q(x) - p(x)q'(x)}{(q(x))^2}.
  2. Apply the quotient rule: Apply the quotient rule using the given values.\newlineWe have p(8)=4p(8) = 4, p′(8)=2p'(8) = 2, q(8)=3q(8) = 3, and q′(8)=−1q'(8) = -1. Plugging these into the quotient rule, we get:\newlinev′(8)=p′(8)q(8)−p(8)q′(8)(q(8))2v'(8) = \frac{p'(8)q(8) - p(8)q'(8)}{(q(8))^2}\newlinev′(8)=2⋅3−4⋅(−1)32v'(8) = \frac{2 \cdot 3 - 4 \cdot (-1)}{3^2}
  3. Perform the calculations: Perform the calculations.\newlinev′(8)=6+432v'(8) = \frac{6 + 4}{3^2}\newlinev′(8)=109v'(8) = \frac{10}{9}

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