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The functions p(x) p(x) and q(x) q(x) are differentiable. \newlineThe function r(x) r(x) is defined as: r(x)=p(x)q(x) r(x)= \frac{p(x)}{q(x)} \newlineIf p(3)=5 p(3)= 5 , p′(3)=2 p'(3)= 2 , q(3)=7 q(3)= 7 , and q′(3)=−3 q'(3)= -3 , what is r′(3) r'(3) ? \newlineSimplify any fractions. \newliner′(3)= r'(3)= _____

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Q. The functions p(x) p(x) and q(x) q(x) are differentiable. \newlineThe function r(x) r(x) is defined as: r(x)=p(x)q(x) r(x)= \frac{p(x)}{q(x)} \newlineIf p(3)=5 p(3)= 5 , p′(3)=2 p'(3)= 2 , q(3)=7 q(3)= 7 , and q′(3)=−3 q'(3)= -3 , what is r′(3) r'(3) ? \newlineSimplify any fractions. \newliner′(3)= r'(3)= _____
  1. Identify Rule: Identify the rule for differentiating a quotient.\newlineThe quotient rule states that if r(x)=p(x)q(x)r(x) = \frac{p(x)}{q(x)}, then r′(x)=p′(x)q(x)−p(x)q′(x)(q(x))2r'(x) = \frac{p'(x)q(x) - p(x)q'(x)}{(q(x))^2}.
  2. Apply Quotient Rule: Apply the quotient rule to find r′(3)r'(3). Using the values given: p(3)=5p(3) = 5, p′(3)=2p'(3) = 2, q(3)=7q(3) = 7, and q′(3)=−3q'(3) = -3, we can substitute these into the quotient rule formula. r′(3)=p′(3)q(3)−p(3)q′(3)(q(3))2r'(3) = \frac{p'(3)q(3) - p(3)q'(3)}{(q(3))^2} r′(3)=2⋅7−5⋅(−3)72r'(3) = \frac{2\cdot7 - 5\cdot(-3)}{7^2}
  3. Perform Calculations: Perform the calculations.\newliner′(3)=14+1549r'(3) = \frac{14 + 15}{49}\newliner′(3)=2949r'(3) = \frac{29}{49}

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