1. Let's start by stating the problem: What does it mean for a function to be differentiable? 2. A function $f(x)$ is said to be differentiable at a point $x=a$ if the derivative $f'(a)$ exists. 3. The derivative at $x=a$ is defined as the limit: $$ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $$ 4. This means the function has a well-defined tangent line at $x=a$, and the slope of this tangent line is $f'(a)$. 5. Important rules: - Differentiability implies continuity at that point, but continuity does not always imply differentiability. - If the limit above does not exist or is infinite, the function is not differentiable at $x=a$. 6. In simpler terms, being differentiable means the function's graph is "smooth" at that point without any sharp corners or cusps. 7. For example, the function $f(x) = |x|$ is continuous everywhere but not differentiable at $x=0$ because the slope from the left and right differ. 8. To check differentiability, you can compute the limit of the difference quotient or check if the left-hand and right-hand derivatives are equal. 9. Summary: Differentiability at a point means the function has a unique, finite slope (derivative) there, allowing us to approximate the function linearly near that point.