1. **Problem statement:** Find the derivative $f'(x)$ for $x < 1$ where the function is defined as $$f(x) = x^2 \text{ for } x \leq 1.$$ 2. **Formula used:** The derivative of a function $f(x)$ is given by $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$$ 3. **Derivative of $x^2$:** Using the power rule, which states $$\frac{d}{dx} x^n = n x^{n-1},$$ for $n=2$, we get $$f'(x) = 2x.$$ 4. **Apply to $x < 1$:** Since the function for $x < 1$ is $x^2$, the derivative for $x < 1$ is $$f'(x) = 2x.$$ **Final answer:** $$\boxed{f'(x) = 2x \text{ for } x < 1}.$$