Find `(dy)/(dx)` if `y=log{e^x((x-2)/(x+2))^(3/4)}`

Explanation

To find \(\frac{dy}{dx}\) for the function

\[ y = \log\left(e^x \left(\frac{x-2}{x+2}\right)^{\frac{3}{4}}\right), \]

we can follow these steps:

### Step 1: Simplify the logarithmic expression
Using the properties of logarithms, we can break down the expression:

\[ y = \log(e^x) + \log\left(\left(\frac{x-2}{x+2}\right)^{\frac{3}{4}}\right). \]

### Step 2: Apply logarithmic identities
Using the identity \(\log(a^b) = b \log(a)\):

\[ y = x + \frac{3}{4} \log\left(\frac{x-2}{x+2}\right). \]

### Step 3: Differentiate \(y\) with respect to \(x\)
Now, we differentiate \(y\):

\[ \frac{dy}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}\left(\frac{3}{4} \log\left(\frac{x-2}{x+2}\right)\right). \]

The derivative of \(x\) is \(1\). For the second term, we will use the chain rule and the quotient rule.

### Step 4: Differentiate the logarithmic term
Using the quotient rule for \(\log\left(\frac{u}{v}\right)\):

\[ \frac{d}{dx}\left(\log\left(\frac{x-2}{x+2}\right)\right) = \frac{1}{\frac{x-2}{x+2}} \cdot \frac{d}{dx}\left(\frac{x-2}{x+2}\right). \]

Now, applying the quotient rule to \(\frac{x-2}{x+2}\):

\[ \frac{d}{dx}\left(\frac{x-2}{x+2}\right) = \frac{(x+2)(1) - (x-2)(1)}{(x+2)^2} = \frac{x+2 - x + 2}{(x+2)^2} = \frac{4}{(x+2)^2}. \]

### Step 5: Substitute back into the derivative
Now substituting back:

\[ \frac{d}{dx}\left(\log\left(\frac{x-2}{x+2}\right)\right) = \frac{(x+2)}{(x-2)} \cdot \frac{4}{(x+2)^2} = \frac{4}{(x-2)(x+2)}. \]

### Step 6: Combine the results
Now, we can combine the results:

\[ \frac{dy}{dx} = 1 + \frac{3}{4} \cdot \frac{4}{(x-2)(x+2)} = 1 + \frac{3}{(x-2)(x+2)}. \]

### Step 7: Simplify the expression
The expression can be simplified further:

\[ \frac{dy}{dx} = 1 + \frac{3}{x^2 - 4}. \]

### Final Result
Thus, the final answer is:

\[ \frac{dy}{dx} = \frac{x^2 - 4 + 3}{x^2 - 4} = \frac{x^2 - 1}{x^2 - 4}. \]

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