How To Know If A Function Is Differentiable

Differentiability is a cornerstone concept in calculus, representing the smoothness of a function at a particular point. Understanding whether a function is differentiable is crucial for various applications, from optimization problems to advanced mathematical modeling. A function that is differentiable at a point possesses a well-defined tangent line, signifying that the function is locally linear around that point.

Prerequisites for Differentiability

Before diving into the methods for determining differentiability, it’s important to understand the concepts upon which it builds:

• Functions: A function is a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

• Limits: A limit describes the value that a function approaches as the input approaches some value.

• Continuity: A function is continuous at a point if it has no breaks, jumps, or holes at that point. Formally, a function f is continuous at a point c if:

  1. f(c) is defined.
  2. The limit of f(x) as x approaches c exists.
  3. The limit of f(x) as x approaches c is equal to f(c).

• Derivatives: The derivative of a function measures the instantaneous rate of change of the function, which is geometrically interpreted as the slope of the tangent line at a given point.

Criteria for Differentiability

A function f(x) is differentiable at a point x = a if the following conditions are met:

1. The function must be continuous at x = a. Continuity is a necessary but not sufficient condition for differentiability. A function can be continuous at a point but not differentiable there.

2. The limit defining the derivative must exist. This means:

   • The left-hand limit of the difference quotient exists:

     lim (h->0-) [f(a + h) - f(a)] / h

   • The right-hand limit of the difference quotient exists:

     lim (h->0+) [f(a + h) - f(a)] / h

   • The left-hand limit and the right-hand limit must be equal:

     lim (h->0-) [f(a + h) - f(a)] / h = lim (h->0+) [f(a + h) - f(a)] / h

3. The derivative must be finite. The derivative cannot tend to infinity at the point x = a.

Methods to Determine Differentiability

To determine if a function is differentiable at a point, several approaches can be employed. These methods often involve checking continuity, evaluating limits, and analyzing the behavior of the function around the point in question.

1. Checking Continuity

As mentioned earlier, continuity is a prerequisite for differentiability. Thus, the first step in determining differentiability is to check if the function is continuous at the point of interest. If the function is not continuous, it cannot be differentiable.

Example:

Consider the function:

f(x) = { x^2, if x < 1

{ 2x - 1, if x >= 1

To check for continuity at x = 1:

1. f(1) = 2(1) - 1 = 1

2. The left-hand limit as x approaches 1:

   lim (x->1-) f(x) = lim (x->1-) x^2 = 1

3. The right-hand limit as x approaches 1:

   lim (x->1+) f(x) = lim (x->1+) (2x - 1) = 1

Since f(1) and both limits are equal to 1, the function is continuous at x = 1.

2. Evaluating Left-Hand and Right-Hand Limits of the Difference Quotient

If the function is continuous at x = a, the next step is to check if the derivative exists at that point. This involves calculating and comparing the left-hand and right-hand limits of the difference quotient.

Example (continued):

Now let's check for differentiability at x = 1.

1. The left-hand limit of the difference quotient:

   lim (h->0-) [f(1 + h) - f(1)] / h = lim (h->0-) [(1 + h)^2 - 1] / h = lim (h->0-) [1 + 2h + h^2 - 1] / h = lim (h->0-) [2h + h^2] / h = lim (h->0-) (2 + h) = 2

2. The right-hand limit of the difference quotient:

   lim (h->0+) [f(1 + h) - f(1)] / h = lim (h->0+) [2(1 + h) - 1 - 1] / h = lim (h->0+) [2 + 2h - 2] / h = lim (h->0+) 2h / h = 2

Since both limits are equal (2), the function is differentiable at x = 1.

3. Identifying Points of Non-Differentiability

Certain features of a function can cause it to be non-differentiable at specific points. These include:

• Corners: A corner occurs when the left-hand and right-hand derivatives exist but are not equal. At a corner, the tangent line is not uniquely defined.
• Cusps: A cusp is similar to a corner, but the function approaches the point with an infinite slope on one or both sides.
• Vertical Tangents: At a point where the tangent is vertical, the derivative is undefined because the slope is infinite.
• Discontinuities: If a function has a discontinuity (jump, removable, or infinite) at a point, it is not differentiable at that point.

Examples:

1. Corner: Consider the function f(x) = |x| at x = 0.

   • For x < 0, f(x) = -x, so f'(x) = -1.
   • For x > 0, f(x) = x, so f'(x) = 1.

   Since the left-hand derivative is -1 and the right-hand derivative is 1, the function has a corner at x = 0 and is not differentiable there.

2. Cusp: Consider the function f(x) = x^(2/3) at x = 0.

   • The derivative f'(x) = (2/3)x^(-1/3) = 2 / (3 * x^(1/3)).

   As x approaches 0, the derivative approaches infinity, creating a cusp at x = 0. Thus, the function is not differentiable at x = 0.

3. Vertical Tangent: Consider the function f(x) = x^(1/3) at x = 0.

   • The derivative f'(x) = (1/3)x^(-2/3) = 1 / (3 * x^(2/3)).

   As x approaches 0, the derivative approaches infinity, indicating a vertical tangent at x = 0. Thus, the function is not differentiable at x = 0.

4. Discontinuity: Consider the Heaviside step function:

   H(x) = { 0, if x < 0

   { 1, if x >= 0

   This function has a jump discontinuity at x = 0 and is not differentiable there.

4. Using Derivative Rules

For many common functions, derivative rules can be applied directly to find the derivative. If the derivative exists and is finite at a point, the function is differentiable at that point.

Example:

Consider the function f(x) = x^3 + 2x^2 - x + 5.

1. Apply the power rule to find the derivative:

   f'(x) = 3x^2 + 4x - 1

Since the derivative exists for all real numbers x, the function is differentiable for all x.

5. Piecewise Functions

Piecewise functions require special attention because their differentiability depends on how the different pieces connect. To determine differentiability at the points where the function changes definition (breakpoints), you must:

1. Check for continuity at the breakpoint.
2. Calculate the left-hand and right-hand derivatives at the breakpoint.
3. Compare the left-hand and right-hand derivatives. If they are equal, the function is differentiable at the breakpoint.

Example:

Consider the function:

f(x) = { x^2, if x <= 2

{ 4x - 4, if x > 2

1. Check for continuity at x = 2:

   • f(2) = 2^2 = 4
   • Left-hand limit: lim (x->2-) f(x) = lim (x->2-) x^2 = 4
   • Right-hand limit: lim (x->2+) f(x) = lim (x->2+) (4x - 4) = 4

   The function is continuous at x = 2.

2. Calculate the left-hand and right-hand derivatives:

   • For x <= 2, f'(x) = 2x
   • For x > 2, f'(x) = 4

3. Evaluate the derivatives at x = 2:

   • Left-hand derivative: f'(2-) = 2(2) = 4
   • Right-hand derivative: f'(2+) = 4

Since both derivatives are equal, the function is differentiable at x = 2.

Common Functions and Differentiability

Knowing the differentiability properties of common functions can save time and effort in many cases.

• Polynomials: Polynomials are differentiable everywhere.
• Rational Functions: Rational functions (ratios of polynomials) are differentiable everywhere except where the denominator is zero.
• Trigonometric Functions: sin(x) and cos(x) are differentiable everywhere. tan(x) is differentiable everywhere except at x = (2n+1)π/2 for integer n, where it has vertical asymptotes.
• Exponential Functions: e^x and a^x (where a > 0) are differentiable everywhere.
• Logarithmic Functions: ln(x) is differentiable for x > 0.
• Root Functions: √x is differentiable for x > 0, but not differentiable at x = 0 due to a vertical tangent.

Advanced Topics in Differentiability

Higher-Order Derivatives

If a function f(x) is differentiable, its derivative f'(x) may also be differentiable. The derivative of the derivative is called the second derivative, denoted as f''(x). Higher-order derivatives can be found by repeatedly differentiating the previous derivative. The existence of higher-order derivatives provides more information about the function's behavior, such as concavity (determined by the second derivative) and rates of change of rates of change.

Smoothness

A function is said to be smooth if it has derivatives of all orders at every point in its domain. Polynomials, sin(x), cos(x), and e^x are examples of smooth functions. Smoothness is a stronger condition than differentiability, as it requires not just the existence of a first derivative, but the existence of derivatives of all orders.

Complex Differentiability

In complex analysis, the concept of differentiability is more stringent than in real analysis. A complex function f(z) is differentiable at a point z = a if the limit:

lim (h->0) [f(a + h) - f(a)] / h

exists, where h is a complex number. This limit must exist regardless of how h approaches 0 in the complex plane. Complex differentiability leads to the concept of analytic functions, which have many desirable properties, such as infinite differentiability and representation by power series.

Applications of Differentiability

The concept of differentiability is fundamental in many areas of mathematics, science, and engineering. Some key applications include:

• Optimization: Finding maximum and minimum values of functions, which is essential in economics, physics, and engineering.
• Curve Sketching: Analyzing the shape of a function by examining its first and second derivatives.
• Related Rates: Determining how the rates of change of different variables are related to each other.
• Taylor Series: Approximating functions using polynomials, which is crucial in numerical analysis and computer science.
• Differential Equations: Modeling physical systems using equations involving derivatives.
• Physics: Describing motion, forces, and other physical phenomena.
• Engineering: Designing structures, circuits, and control systems.

Examples and Case Studies

Example 1: Analyzing a Piecewise Function

Consider the function:

f(x) = { x^2 + 1, if x < 0

{ 1 - x, if x >= 0

1. Check for continuity at x = 0:

   • f(0) = 1 - 0 = 1
   • Left-hand limit: lim (x->0-) f(x) = lim (x->0-) (x^2 + 1) = 1
   • Right-hand limit: lim (x->0+) f(x) = lim (x->0+) (1 - x) = 1

   The function is continuous at x = 0.

2. Calculate the derivatives:

   • For x < 0, f'(x) = 2x
   • For x >= 0, f'(x) = -1

3. Evaluate the derivatives at x = 0:

   • Left-hand derivative: f'(0-) = 2(0) = 0
   • Right-hand derivative: f'(0+) = -1

Since the left-hand derivative (0) is not equal to the right-hand derivative (-1), the function is not differentiable at x = 0.

Example 2: A Function with a Vertical Tangent

Consider the function f(x) = (x - 1)^(1/3).

1. Find the derivative:

   f'(x) = (1/3)(x - 1)^(-2/3) = 1 / (3 * (x - 1)^(2/3))

2. Analyze the derivative at x = 1:

As x approaches 1, the derivative approaches infinity, indicating a vertical tangent at x = 1. Thus, the function is not differentiable at x = 1.

Example 3: A Polynomial Function

Consider the function f(x) = 5x^4 - 3x^2 + 7x - 2.

1. Find the derivative:

   f'(x) = 20x^3 - 6x + 7

Since the derivative exists for all real numbers x, the function is differentiable for all x. Furthermore, higher-order derivatives can be found:

• f''(x) = 60x^2 - 6
• f'''(x) = 120x
• f''''(x) = 120

All derivatives exist, so the function is smooth.

Practical Tips and Tricks

• Visualize the function: Graphing the function can help identify potential points of non-differentiability, such as corners, cusps, and discontinuities.
• Memorize common derivatives: Knowing the derivatives of common functions can speed up the process of finding derivatives.
• Use derivative rules: Applying derivative rules correctly is crucial for finding derivatives accurately.
• Check for continuity first: Always check for continuity before attempting to find the derivative.
• Pay attention to piecewise functions: Piecewise functions require special care at breakpoints to ensure differentiability.
• Practice, practice, practice: The more you practice finding derivatives, the better you will become at identifying differentiable functions.

Conclusion

Understanding differentiability is essential for calculus and its applications. By mastering the criteria and methods for determining differentiability, you can analyze the behavior of functions, solve optimization problems, and model real-world phenomena. Remember to check for continuity, evaluate limits, identify points of non-differentiability, and apply derivative rules correctly. With practice and a solid understanding of the concepts, you can confidently determine whether a function is differentiable at any given point.