To find the derivative of the function $f(x) = \frac{1}{3\sqrt{x^2}}$, follow these steps:

Step 1: Simplify the expression

Rewrite the function as: $f(x) = \frac{1}{3} \cdot (x^2)^{-\frac{1}{2}}$ This is because $\sqrt{x^2} = (x^2)^{1/2}$, and taking the reciprocal turns the exponent into $-\frac{1}{2}$.

Step 2: Apply the power rule for derivatives

The power rule states that if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.

For $f(x) = \frac{1}{3} \cdot x^{-1}$, we can differentiate it as follows: $f'(x) = \frac{1}{3} \cdot (-\frac{1}{2}) \cdot x^{-\frac{3}{2}} = -\frac{1}{6} \cdot x^{-\frac{3}{2}}$

Thus, the derivative is: $f'(x) = -\frac{1}{6x^{3/2}}$

Final Answer:

$f'(x) = -\frac{1}{6x^{3/2}}$

Would you like further explanation or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the chain rule and how could it apply to derivatives like this?
2. How do you differentiate higher powers of $x$ or other polynomial functions?
3. What happens when the function involves a square root or a cube root instead of just a square?
4. Can you explain the product rule and how it differs from the power rule?
5. How would you handle a situation where both the numerator and denominator contain powers of $x$?

Tip: When differentiating functions with roots, it's often helpful to rewrite them with exponents to make the differentiation process easier!