Questions: Use the Generalized Power Rule to find the derivative of the function. f(x)=1/(sqrt(5 x^2-5 x+8)) f'(x)=

Transcript text: Use the Generalized Power Rule to find the derivative of the function. \[ \begin{array}{l} f(x)=\frac{1}{\sqrt{5 x^{2}-5 x+8}} \\ f^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

Step 1: Define the Function

We start with the function given by

\[ f(x) = \frac{1}{\sqrt{5x^2 - 5x + 8}}. \]

Step 2: Rewrite the Function

To apply the Generalized Power Rule, we rewrite the function using a negative fractional exponent:

\[ f(x) = (5x^2 - 5x + 8)^{-\frac{1}{2}}. \]

Step 3: Apply the Chain Rule

Using the chain rule, we differentiate \( f(x) \):

\[ f'(x) = -\frac{1}{2}(5x^2 - 5x + 8)^{-\frac{3}{2}} \cdot (10x - 5). \]

Step 4: Simplify the Derivative

Now, we simplify the expression for \( f'(x) \):

\[ f'(x) = -\frac{(10x - 5)}{2(5x^2 - 5x + 8)^{\frac{3}{2}}}. \]

Final Answer

Thus, the derivative of the function is

\[ \boxed{f'(x) = \frac{5 - 10x}{2(5x^2 - 5x + 8)^{\frac{3}{2}}}}. \]

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