1. Let's start by stating the problem: We want to understand what a differentiable function is and how to determine if a function is differentiable. 2. A function $f(x)$ is said to be differentiable at a point $x=a$ if the derivative $f'(a)$ exists. This means the function has a well-defined tangent line at that point. 3. The derivative at $x=a$ is defined by the limit: $$ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $$ If this limit exists (is finite), then $f$ is differentiable at $a$. 4. Important rules: - Differentiability implies continuity, but continuity does not necessarily imply differentiability. - If a function has a sharp corner or cusp at $a$, it is not differentiable there. - If the function is differentiable on an interval, it means the derivative exists at every point in that interval. 5. To check differentiability: - Compute the limit definition of the derivative. - Alternatively, if the function is composed of elementary functions (polynomials, exponentials, trigonometric, etc.) combined by addition, multiplication, division (where denominator is not zero), it is differentiable where it is defined. 6. Example: Consider $f(x) = |x|$ at $x=0$. - Compute left-hand derivative: $$ \lim_{h \to 0^-} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1 $$ - Compute right-hand derivative: $$ \lim_{h \to 0^+} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1 $$ - Since left and right derivatives are not equal, $f$ is not differentiable at $0$. 7. Summary: Differentiability means the function's rate of change is well-defined at a point, which requires the limit defining the derivative to exist and be the same from both sides. Final answer: A function is differentiable at a point if its derivative exists there, meaning the limit of the difference quotient exists and is finite.