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The functions f(x) f(x) and g(x) g(x) are differentiable. \newlineThe function h(x) h(x) is defined as: h(x)=g(x)f(x) h(x)=\frac{g(x)}{f(x)} \newlineIf f(8)=1 f(8)=1 , f′(8)=−6 f'(8)=-6 , g(8)=8 g(8)=8 , and g′(8)=−10 g'(8)=-10 , what is h′(8) h'(8) ? \newlineSimplify any fractions. \newlineh′(8)= h'(8)= ______

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Q. The functions f(x) f(x) and g(x) g(x) are differentiable. \newlineThe function h(x) h(x) is defined as: h(x)=g(x)f(x) h(x)=\frac{g(x)}{f(x)} \newlineIf f(8)=1 f(8)=1 , f′(8)=−6 f'(8)=-6 , g(8)=8 g(8)=8 , and g′(8)=−10 g'(8)=-10 , what is h′(8) h'(8) ? \newlineSimplify any fractions. \newlineh′(8)= h'(8)= ______
  1. Apply Quotient Rule: Apply the quotient rule to find h′(x)h'(x). The quotient rule states that if h(x)=g(x)f(x)h(x) = \frac{g(x)}{f(x)}, then h′(x)=f(x)g′(x)−g(x)f′(x)(f(x))2h'(x) = \frac{f(x)g'(x) - g(x)f'(x)}{(f(x))^2}.
  2. Substitute Given Values: Substitute the given values into the quotient rule formula.\newlineWe have f(8)=1f(8) = 1, f′(8)=−6f'(8) = -6, g(8)=8g(8) = 8, and g′(8)=−10g'(8) = -10.\newlineSo, h′(8)=(1×−10−8×−6)/(1)2h'(8) = (1 \times -10 - 8 \times -6) / (1)^2.
  3. Perform Calculations: Perform the calculations.\newlineh′(8)=(−10+48)/1.h'(8) = (-10 + 48) / 1.\newlineh′(8)=38/1.h'(8) = 38 / 1.\newlineh′(8)=38.h'(8) = 38.

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