To derive $f\left(\frac{1}{x^2}\right)$, let's assume $f(u)$ is a differentiable function where $u = \frac{1}{x^2}$. We'll use the chain rule for differentiation.

Steps:

1. Set $u = \frac{1}{x^2}$: $\frac{du}{dx} = \frac{d}{dx}\left(x^{-2}\right) = -2x^{-3} = -\frac{2}{x^3}.$

2. Apply the chain rule: $\frac{d}{dx}f\left(\frac{1}{x^2}\right) = f'(u) \cdot \frac{du}{dx}.$

3. Substitute $u$ and $\frac{du}{dx}$: $\frac{d}{dx}f\left(\frac{1}{x^2}\right) = f'\left(\frac{1}{x^2}\right) \cdot \left(-\frac{2}{x^3}\right).$

Final Answer:

$\frac{d}{dx}f\left(\frac{1}{x^2}\right) = -\frac{2}{x^3} \cdot f'\left(\frac{1}{x^2}\right).$

Would you like a more detailed explanation of the chain rule? Let me know! 😊

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Related Questions:

1. How does the chain rule work in differentiation?
2. What is the derivative of $g(x) = \frac{1}{x^2}$?
3. Can this result be used in implicit differentiation? How?
4. What is the second derivative of $f\left(\frac{1}{x^2}\right)$?
5. How does this apply to real-world problems involving rates of change?

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Tip:

Always substitute intermediate variables (like $u$) carefully to simplify the differentiation process when using the chain rule!