To differentiate the function $f(x) = x \cdot e^{-2x}$, we need to apply the product rule, since it's a product of two functions: $x$ and $e^{-2x}$.

The product rule states that: $(f \cdot g)' = f' \cdot g + f \cdot g'$ where $f(x) = x$ and $g(x) = e^{-2x}$.

1. Differentiate $f(x) = x$: $f'(x) = 1$

2. Differentiate $g(x) = e^{-2x}$. Using the chain rule: $g'(x) = e^{-2x} \cdot (-2) = -2e^{-2x}$

Now, applying the product rule: $\frac{d}{dx} \left( x \cdot e^{-2x} \right) = 1 \cdot e^{-2x} + x \cdot (-2e^{-2x})$ Simplifying: $\frac{d}{dx} \left( x \cdot e^{-2x} \right) = e^{-2x} - 2x \cdot e^{-2x}$ Thus, the derivative is: $f'(x) = e^{-2x}(1 - 2x)$

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Here are 5 questions related to this topic:

1. How is the chain rule applied in differentiation?
2. What is the product rule, and when do we use it?
3. How do you simplify expressions involving exponentials after applying the product rule?
4. Can you differentiate more complex functions like $x^2 e^{-2x}$?
5. How do you compute second derivatives using the product and chain rules?

Tip: Always simplify the expression after applying the product or chain rule to make the derivative clearer and easier to interpret.