1. The problem is to understand the concept of derivability (differentiability) of a function. 2. Derivability means that a function has a derivative at a given point, which represents the slope of the tangent line to the function at that point. 3. The derivative of a function $f$ at a point $x=a$ is defined by the limit: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ 4. For a function to be derivable at $a$, this limit must exist and be finite. 5. Important rules: - If $f$ is differentiable at $a$, then $f$ is continuous at $a$. - The converse is not always true: continuity does not imply differentiability. 6. To check derivability, compute the limit above or use known derivative rules if the function is standard. 7. Example: For $f(x) = x^2$, the derivative at $a$ is: $$f'(a) = \lim_{h \to 0} \frac{(a+h)^2 - a^2}{h} = \lim_{h \to 0} \frac{a^2 + 2ah + h^2 - a^2}{h} = \lim_{h \to 0} \frac{2ah + h^2}{h}$$ 8. Simplify by canceling $h$: $$= \lim_{h \to 0} (2a + h) = 2a$$ 9. So, $f$ is derivable everywhere and $f'(x) = 2x$. This explains the concept and how to check derivability.