If `f(x) = log _(x^(2) ) x^(3) ` write the value of `f'(x)`

Answer

To find the derivative \( f'(x) \) of the function \( f(x) = \log_{x^2}(x^3) \), we can follow these steps:

### Step 1: Rewrite the logarithm using the change of base formula
The change of base formula for logarithms states that:
\[ \log_a b = \frac{\log b}{\log a} \]
Applying this to our function, we have:
\[ f(x) = \log_{x^2}(x^3) = \frac{\log(x^3)}{\log(x^2)} \]

### Step 2: Simplify the logarithmic expressions
Using the property of logarithms that states \( \log(a^b) = b \log(a) \), we can simplify both the numerator and the denominator:
\[ f(x) = \frac{3 \log(x)}{2 \log(x)} \]

### Step 3: Cancel the common terms
Since \( \log(x) \) is in both the numerator and the denominator (assuming \( x > 0 \) and \( x \neq 1 \)), we can cancel it:
\[ f(x) = \frac{3}{2} \]

### Step 4: Differentiate \( f(x) \)
Now that we have \( f(x) = \frac{3}{2} \), we can find the derivative:
\[ f'(x) = 0 \]
because the derivative of a constant is zero.

### Conclusion
Thus, the value of \( f'(x) \) is:
\[ \boxed{0} \]