To evaluate \( \frac{f}{g}'(8) \), we will use the quotient rule of differentiation. The quotient rule states that if you have two functions \( f(x) \) and \( g(x) \), the derivative of their quotient is given by:
\[ \left(\frac{f}{g}\right)' = \frac{f' \cdot g - f \cdot g'}{g^2} \]
Given the values:
- \( f(8) = 1 \)
- \( f'(8) = -2 \)
- \( g(8) = \frac{1}{5} \)
- \( g'(8) = 2 \)
We will plug these values into the quotient rule formula.
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Calculate \( f'(8) \cdot g(8) \): \[ f'(8) \cdot g(8) = -2 \cdot \frac{1}{5} = -\frac{2}{5} \]
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Calculate \( f(8) \cdot g'(8) \): \[ f(8) \cdot g'(8) = 1 \cdot 2 = 2 \]
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Substitute these results back into the quotient rule formula: \[ \left( \frac{f}{g} \right)'(8) = \frac{-\frac{2}{5} - 2}{\left(\frac{1}{5}\right)^2} \]
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Calculate the denominator: \[ \left(\frac{1}{5}\right)^2 = \frac{1}{25} \]
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Now calculate the numerator: \[ -\frac{2}{5} - 2 = -\frac{2}{5} - \frac{10}{5} = -\frac{12}{5} \]
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Putting it all together: \[ \left( \frac{f}{g} \right)'(8) = \frac{-\frac{12}{5}}{\frac{1}{25}} = -\frac{12}{5} \cdot 25 = -60 \]
So, the final result is: \[ \frac{f}{g}'(8) = -60 \]