What Is Differentiability in Calculus?

Differentiability is a foundational concept in calculus that provides a rigorous criterion for when a function possesses a well-defined tangent at each point in its domain. It extends the concept of continuity by demanding a specific limiting behavior of the function's increments, directly leading to the existence of derivatives.

Formal Definition of Differentiability at a Point

Let $f : D \to \mathbb{R}$ be a function, and let $c \in D$ be an interior point of $D$. The function $f$ is said to be differentiable at $c$ if the following limit exists and is finite:

\[ \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \] The value of this limit, when it exists, is denoted by $f'(c)$ and is called the derivative of $f$ at $c$.

Algebraic Expression for Differentiability

A function $f$ is differentiable at $x = c$ if and only if the left-hand limit (LHL) and right-hand limit (RHL) of the difference quotient at $c$ exist and are equal:

\[ \text{LHL} = \lim_{h \to 0^-} \frac{f(c + h) - f(c)}{h} \] \[ \text{RHL} = \lim_{h \to 0^+} \frac{f(c + h) - f(c)}{h} \] \[ \text{If}~\text{LHL} = \text{RHL} = L~\text{(finite)},~f~\text{is differentiable at}~x = c~\text{with}~f'(c) = L. \]

Relationship between Differentiability and Continuity

If a function $f$ is differentiable at $x = c$, then $f$ is necessarily continuous at $x = c$. However, continuity alone does not guarantee differentiability. This establishes the strict hierarchy: Differentiability implies continuity, but not vice versa. A function may be continuous at a point, yet not differentiable there.

Differentiability Criteria and Continuity Check

To verify differentiability at $x = c$, it is essential first to check the continuity of $f$ at $x = c$:

\[ \lim_{x \to c} f(x) = f(c). \] Without continuity at $x = c$, the differentiability does not arise at that point.

Key Theorem: Differentiability Implies Continuity

Theorem: If $f$ is differentiable at $x = c$, then $f$ is continuous at $x = c$.

Proof:

Suppose $f$ is differentiable at $c$. By definition,

\[ \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} = L~\text{(exists and is finite)} \]

Consider $\lim_{h \to 0} f(c + h)$. Rewrite:

\[ f(c + h) = f(c) + [f(c + h) - f(c)] \]

So,

\[ \lim_{h \to 0} f(c + h) = f(c) + \lim_{h \to 0} [f(c + h) - f(c)] \]

Now, observe:

\[ \lim_{h \to 0} [f(c + h) - f(c)] = \lim_{h \to 0} \left( \frac{f(c + h) - f(c)}{h} \cdot h \right) \]

Because $\lim_{h \to 0} \frac{f(c + h) - f(c)}{h} = L$ and $\lim_{h \to 0} h = 0$, it follows from limit properties that

\[ \lim_{h \to 0} [f(c + h) - f(c)] = L \times 0 = 0 \]

Hence,

\[ \lim_{h \to 0} f(c + h) = f(c) \]

Conclusion: $f$ is continuous at $x = c$.

Differentiability Test and Analytical Approach

For a function $f(x)$ at $x = c$, the differentiability test is performed by explicitly calculating the left-hand and right-hand derivatives:

\[ f'_-(c) = \lim_{h \to 0^-} \frac{f(c + h) - f(c)}{h} \] \[ f'_+(c) = \lim_{h \to 0^+} \frac{f(c + h) - f(c)}{h} \] If $f'_-(c)$ and $f'_+(c)$ are both finite and equal, $f$ is differentiable at $x = c$. Otherwise, $f$ is not differentiable at $x = c$.

The above approach must be executed with each step explicit. This is crucial for points where the formula for $f(x)$ changes, such as at endpoints or in piecewise definitions.

Basic Rules of Differentiability for Single Variable Functions

If $f$ and $g$ are differentiable at $x = c$, and $\alpha, \beta \in \mathbb{R}$, the following differentiability rules hold:

\[ \text{(i)~Sum Rule:}~(f + g)'(c) = f'(c) + g'(c) \] \[ \text{(ii)~Difference Rule:}~(f - g)'(c) = f'(c) - g'(c) \] \[ \text{(iii)~Constant Multiple:}~(\alpha f)'(c) = \alpha f'(c) \] \[ \text{(iv)~Product Rule:}~(fg)'(c) = f'(c)g(c) + f(c)g'(c) \] \[ \text{(v)~Quotient Rule:}~\left( \frac{f}{g} \right)'(c) = \frac{f'(c)g(c) - f(c)g'(c)}{[g(c)]^2},~\text{provided}~g(c) \neq 0 \]

These rules are proven using the limit definition of the derivative and are universal for all differentiable functions.

For advanced applications to composite functions, see the page on Differentiability of Composite Functions.

Differentiability and Graphical Interpretation

On the graph of a function $y = f(x)$, differentiability at a point $x = c$ corresponds to the existence of a unique, non-vertical tangent at $(c, f(c))$. Common signs of non-differentiability include:

- Sharp corners or cusps (e.g., $f(x) = |x|$ at $x=0$) - Vertical tangents (e.g., $f(x) = \sqrt[3]{x}$ at $x=0$) - Discontinuities (points where $f$ is not continuous)

Further geometric connections between limit, continuity, and differentiability can be reviewed at Limit, Continuity, and Differentiability.

Example: Differentiability of the Absolute Value Function

Given: $f(x) = |x|$. Examine differentiability at $x = 0$.

Left-hand derivative: \[ f'_-(0) = \lim_{h \to 0^-} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0^-} \frac{|h|}{h} \] For $h < 0$, $|h| = -h$, so \[ = \lim_{h \to 0^-} \frac{-h}{h} = \lim_{h \to 0^-} (-1) = -1 \]

Right-hand derivative: \[ f'_+(0) = \lim_{h \to 0^+} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0^+} \frac{|h|}{h} \] For $h > 0$, $|h| = h$, so \[ = \lim_{h \to 0^+} \frac{h}{h} = \lim_{h \to 0^+} (1) = 1 \]

Final result: Since left-hand and right-hand derivatives at $x=0$ are not equal, $f(x)$ is not differentiable at $x = 0$, though it is continuous there.

Differentiability in Calculus for Piecewise Functions

For piecewise-defined functions, differentiability at the junction point requires:

1. Continuity at the junction point. 2. Equality of derivatives from both sides at that point.

A systematic approach employing left-hand and right-hand limits of the difference quotient is mandatory. This method extends to absolute value, greatest integer, and trigonometric combinations at critical points.

Differentiability in Multivariable Functions

For $f : \mathbb{R}^2 \to \mathbb{R}$, differentiability at $(a, b)$ is defined as follows: There exist real numbers $p$ and $q$ such that \[ \lim_{(h, k) \to (0,0)} \frac{f(a + h, b + k) - f(a, b) - ph - qk}{\sqrt{h^2 + k^2}} = 0 \] Here, $p$ and $q$ represent the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at $(a, b)$. Multivariable differentiability is stricter than the existence of partial derivatives, as it demands a local linear approximation. For additional exploration of higher-order applications, consult the Differential Calculus topic page.

Summary of Differentiability Formulae

The standard differentiability formula for a real function at $x = c$ is \[ f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \] This serves as the basic test and computational method across all single variable differentiability checks.

Sample Problem: Differentiability at an Endpoint

Given: $f(x) = x^2$ defined on $[0, \infty)$. Check differentiability at $x = 0$.

Left-hand derivative: Not defined, since $x < 0$ is not in domain.

Right-hand derivative: \[ \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h^2 - 0}{h} = \lim_{h \to 0^+} h = 0 \]

Final result: Since only the right-hand derivative exists and is finite, $f(x)$ is said to be right-differentiable at $x=0$. Full differentiability only holds at interior points of the domain.

Advanced Connections and Related Concepts

Mastery of differentiability facilitates the study of L'Hospital's Rule for Indeterminate Limits, Taylor expansions, and the formulation and solution of Differential Equations. Each topic builds upon the precise limiting behavior and local linear character provided by differentiability.