9. Find \( f'(0) \) given the following:

\( f(x) = \left( \frac{h(x)}{g(x)} \right)^2 \)






Answer is: option1

\( \frac{56}{125} \)

Solution:

\( f(x) = \left( \frac{h(x)}{g(x)} \right)^2 \)

\( f'(x) = 2 \cdot \frac{h(x)}{g(x)} \cdot \frac{g(x) h'(x) - h(x) g'(x)}{(g(x))^2} \)

\( f'(0) = 2 \cdot \frac{h(0)}{g(0)} \cdot \frac{g(0) h'(0) - h(0) g'(0)}{(g(0))^2} \)

\( f'(0) = 2 \cdot \frac{-2}{5} \cdot \frac{5 \cdot (-4) - (-2) \cdot 3}{(5)^2} \)

\( f'(0) = 2 \cdot \frac{-2}{5} \cdot \frac{-20 + 6}{25} \)

\( f'(0) = 2 \cdot \frac{-2}{5} \cdot \frac{-14}{25} \)

\( f'(0) = \frac{56}{125} \, \text{Ans} \)

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