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The functions g(x) g(x) and h(x) h(x) are differentiable. \newlineThe function i(x) i(x) is defined as: i(x)=g(x)h(x) i(x)= \frac{g(x)}{h(x)} \newlineIf g(8)=4 g(8)= 4 , g′(8)=3 g'(8)= 3 , h(8)=2 h(8)= 2 , and h′(8)=−4 h'(8)= -4 , what is i′(8) i'(8) ? \newlineSimplify any fractions. \newlinei′(8)= i'(8)= _____

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Q. The functions g(x) g(x) and h(x) h(x) are differentiable. \newlineThe function i(x) i(x) is defined as: i(x)=g(x)h(x) i(x)= \frac{g(x)}{h(x)} \newlineIf g(8)=4 g(8)= 4 , g′(8)=3 g'(8)= 3 , h(8)=2 h(8)= 2 , and h′(8)=−4 h'(8)= -4 , what is i′(8) i'(8) ? \newlineSimplify any fractions. \newlinei′(8)= i'(8)= _____
  1. Apply Quotient Rule: Use the quotient rule for differentiation: (uv)′=u′v−uv′v2(\frac{u}{v})^{'} = \frac{u^{'}v - uv^{'}}{v^2}. Here, u=g(x)u = g(x) and v=h(x)v = h(x).
  2. Calculate Derivatives: Calculate u′(8)u'(8) and v′(8)v'(8) using the given derivatives: u′(8)=g′(8)=3u'(8) = g'(8) = 3, v′(8)=h′(8)=−4v'(8) = h'(8) = -4.
  3. Calculate Function Values: Calculate u(8)u(8) and v(8)v(8) using the given function values: u(8)=g(8)=4u(8) = g(8) = 4, v(8)=h(8)=2v(8) = h(8) = 2.
  4. Plug into Formula: Plug the values into the quotient rule formula: i′(8)=(3×2−4×−4)(22)i^{'}(8) = \frac{(3 \times 2 - 4 \times -4)}{(2^2)}.
  5. Simplify Numerator: Simplify the numerator: i′(8)=(6+16)4i^{'}(8) = \frac{(6 + 16)}{4}.
  6. Simplify Fraction: Simplify the fraction: i(8)=224i^{(8)} = \frac{22}{4}.
  7. Reduce to Lowest Terms: Reduce the fraction to lowest terms: i(8)=112i^{(8)} = \frac{11}{2}.

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