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The functions f(x) f(x) and g(x) g(x) are differentiable. The function h(x) h(x) is defined as: h(x)=f(x)g(x) h(x)= \frac{f(x)}{g(x)} If f(9)=6 f(9)= 6 , f′(9)=4 f'(9)= 4 , g(9)=2 g(9)= 2 , and g′(9)=−3 g'(9)= -3 , what is h′(9) h'(9) ? Simplify any fractions. h′(9)= h'(9)=

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Q. The functions f(x) f(x) and g(x) g(x) are differentiable. The function h(x) h(x) is defined as: h(x)=f(x)g(x) h(x)= \frac{f(x)}{g(x)} If f(9)=6 f(9)= 6 , f′(9)=4 f'(9)= 4 , g(9)=2 g(9)= 2 , and g′(9)=−3 g'(9)= -3 , what is h′(9) h'(9) ? Simplify any fractions. h′(9)= h'(9)=
  1. Given function and derivative: We are given the function h(x)=f(x)g(x) h(x) = \frac{f(x)}{g(x)} and we need to find the derivative of h h at x=9 x = 9 , which is h′(9) h'(9) . To do this, we will use the quotient rule for derivatives, which states that if h(x)=f(x)g(x) h(x) = \frac{f(x)}{g(x)} , then h′(x)=g(x)f′(x)−f(x)g′(x)(g(x))2 h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} .
  2. Applying the quotient rule: We are given f(9)=6f(9) = 6, f′(9)=4f'(9) = 4, g(9)=2g(9) = 2, and g′(9)=−3g'(9) = -3. Let's plug these values into the quotient rule formula to find h′(9)h'(9).\newlineh′(9)=g(9)f′(9)−f(9)g′(9)(g(9))2h'(9) = \frac{g(9)f'(9) - f(9)g'(9)}{(g(9))^2}\newlineh′(9)=2⋅4−6⋅(−3)(2)2h'(9) = \frac{2 \cdot 4 - 6 \cdot (-3)}{(2)^2}
  3. Plugging in values: Now we perform the calculations:\newlineh'(99) = 8+184\frac{8 + 18}{4}\newlineh'(99) = 264\frac{26}{4}\newlineh'(99) = 6.56.5

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