Questions: Determine the derivative of the function g(x) = ln(sqrt[3]((x+9)/(x-9))).

Determine the derivative of the function g(x) = ln(sqrt[3]((x+9)/(x-9))).
Transcript text: Determine the derivative of the function $g(x)=\ln \left(\sqrt[3]{\frac{x+9}{x-9}}\right)$.
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Solution

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Solution Steps

Step 1: Simplifying the Function

We start with the function \( g(x) = \ln \left(\sqrt[3]{\frac{x+9}{x-9}}\right) \). Using the property of logarithms, we can rewrite this as: \[ g(x) = \frac{1}{3} \ln \left(\frac{x+9}{x-9}\right) \]

Step 2: Differentiating the Function

Next, we differentiate \( g(x) \) using the chain rule. The derivative is given by: \[ g^{\prime}(x) = \frac{1}{3} \cdot \frac{(x - 9) \cdot \left( \frac{1}{x - 9} - \frac{x + 9}{(x - 9)^2} \right)}{(x + 9)} \]

Step 3: Simplifying the Derivative

After simplification, we find that: \[ g^{\prime}(x) = \frac{-6.0}{x^2 - 81.0} \]

Final Answer

Thus, the derivative of the function is: \[ \boxed{g^{\prime}(x) = \frac{-6.0}{x^2 - 81.0}} \]

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