Determining whether a function is differentiable at a specific point or across an interval is a fundamental question in calculus that probes the smoothness of a curve. While continuity is a prerequisite, it is not sufficient; a function can be continuous yet possess a sharp corner or a vertical tangent that prevents the existence of a unique tangent line. The core of the investigation lies in the behavior of the difference quotient, which serves as the foundation for the analytical tests used to confirm differentiability.
The Foundational Definition: The Limit of the Difference Quotient
At its heart, the question of differentiability is answered by examining the limit that defines the derivative. A function \( f(x) \) is differentiable at a point \( x = a \) if the limit of the difference quotient exists and is finite. This limit, expressed as \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \), must approach the same finite value regardless of whether \( h \) approaches zero from the positive side (right-hand limit) or the negative side (left-hand limit. If this condition is met, the function possesses a well-defined, non-vertical tangent line at that specific coordinate.
Checking for Continuity: The Necessary First Step
Before applying the more nuanced tests, it is efficient to verify that the function is continuous at the point in question, as discontinuities immediately rule out differentiability. A function must satisfy the condition that the limit as \( x \) approaches \( a \) is equal to the function's value at \( a \), written as \( \lim_{x \to a} f(x) = f(a) \). If a function has a jump, an asymptote, or a removable hole at \( x = a \), it is inherently non-differentiable at that location, saving time by avoiding unnecessary calculations.
Identifying Corners and Cusps
Graphical analysis provides immediate visual cues regarding differentiability, primarily through the identification of corners and cusps. At a corner, such as the peak of the absolute value function at \( x = 0 \), the left-hand and right-hand tangents are distinct, resulting in a sudden change in direction. Similarly, a cusp features a point where the slopes on either side approach infinity but with opposite signs. In both scenarios, the difference quotient fails to converge to a single finite value, causing the limit to not exist.
Detecting Vertical Tangents
Another scenario where a function fails to be differentiable is the presence of a vertical tangent line. This occurs when the slope of the curve increases without bound as it approaches a specific point. Mathematically, this is identified when the limit of the derivative approaches positive or negative infinity. While the function is continuous at this point, the infinite slope means the difference quotient grows without bound, violating the requirement for a finite limit and thus disqualifying the point from being differentiable.
Applying Analytical Rules to Common Functions
For standard algebraic functions, specific rules derived from calculus provide a shortcut for determining differentiability without resorting to the limit definition every time. Polynomial functions, which include expressions composed of constants and variables raised to non-negative integer powers, are differentiable everywhere on the real number line. Rational functions, formed as the ratio of two polynomials, are differentiable at every point where the denominator is non-zero, as division by zero creates a discontinuity that breaks the smoothness.
