Continuity and Differentiability - Differentiability at a Point

• In mathematics, differentiability is a property of functions that describes the smoothness of their graphs.
• A function is said to be differentiable at a point if its derivative exists at that point.
• The derivative represents the rate of change of a function at any point.
• It gives us information about the slope or steepest direction of the function’s graph at a particular point.
• Differentiability and continuity are closely related concepts in calculus.

Conditions for Differentiability at a Point

To determine whether a function is differentiable at a particular point, we need to check the following conditions:

1. The function must be defined at that point.

2. The function must be continuous at that point.

3. The derivative of the function must exist at that point.

Differentiability vs. Continuity

• Differentiability implies continuity, but continuity does not guarantee differentiability.
• A function can be continuous without being differentiable.
• However, if a function is differentiable at a point, it must be continuous at that point. Example: Consider the function f(x) = |x| at x = 0.
• The function is not differentiable at x = 0 because the derivative does not exist.
• However, the function is continuous at x = 0.

Geometric Interpretation of Differentiability

• Geometrically, differentiability is related to the smoothness of a graph.
• If a function is differentiable at a point, its graph has a well-defined tangent line at that point.
• This tangent line represents the best linear approximation to the function near that point.
• The function is not differentiable at a point if its graph has a sharp corner, vertical tangent line, or a discontinuity at that point. Example: Consider the function f(x) = √(x^2 - 1).
• The graph of this function has a vertical tangent line at x = 1, which means it is not differentiable at that point.
• However, it is differentiable for all other values of x.

Differentiability of Basic Functions

• Most elementary functions, such as polynomials, trigonometric functions, exponential functions, etc., are differentiable.
• The sum, difference, product, and quotient of differentiable functions are also differentiable.
• The chain rule allows us to differentiate composite functions.
• However, some functions, like absolute value and floor functions, are not differentiable at some points. Equation: d/dx (e^x) = e^x Equation: d/dx (sin x) = cos x Equation: d/dx (cos x) = -sin x Equation: d/dx (x^n) = nx^(n-1), where n is a real number

Example: Differentiability at a Point

Let’s consider the function f(x) = x^2 + 3x - 2.

• The function is polynomial, and hence, it is differentiable for all values of x.
• To check the differentiability at a point, we need to find the derivative of f(x) and evaluate it at that point.
• By differentiating f(x), we get f’(x) = 2x + 3.
• Now, let’s find the derivative at x = 1.
• Substituting x = 1 in f’(x), we get f’(1) = 2(1) + 3 = 2 + 3 = 5.
• Since f’(1) exists, the function f(x) is differentiable at x = 1.

Conditions for Differentiability at a Point

• The function must be defined at that point.
• The function must be continuous at that point.
• The derivative of the function must exist at that point.

  Differentiability vs. Continuity

• Differentiability implies continuity, but continuity does not guarantee differentiability.
• A function can be continuous without being differentiable.

  Geometric Interpretation of Differentiability

• Geometrically, differentiability is related to the smoothness of a graph.
• If a function is differentiable at a point, its graph has a well-defined tangent line at that point.
• The function is not differentiable at a point if its graph has a sharp corner, vertical tangent line, or a discontinuity at that point.

  Differentiability of Basic Functions

• Most elementary functions, such as polynomials, trigonometric functions, exponential functions, etc., are differentiable.
• The sum, difference, product, and quotient of differentiable functions are also differentiable.
• The chain rule allows us to differentiate composite functions.
• However, some functions, like absolute value and floor functions, are not differentiable at some points. Example: Differentiability at a Point
• Let’s consider the function f(x) = x^2 + 3x - 2.
• The function is polynomial, and hence, it is differentiable for all values of x.
• To check the differentiability at a point, we need to find the derivative of f(x) and evaluate it at that point.
• By differentiating f(x), we get f’(x) = 2x + 3.
• Now, let’s find the derivative at x = 1.
• Substituting x = 1 in f’(x), we get f’(1) = 2(1) + 3 = 2 + 3 = 5.
• Since f’(1) exists, the function f(x) is differentiable at x = 1. Differentiability of Absolute Value Function
• The absolute value function, f(x) = |x|, is not differentiable at x = 0.
• The function has a sharp corner at x = 0, which makes the derivative undefined at that point.
• However, the function is differentiable for all other values of x. Differentiability of Piecewise Functions
• Piecewise functions may or may not be differentiable at the points where the pieces join.
• To check the differentiability at a point of discontinuity, we need to evaluate the one-sided derivatives.
• If the one-sided derivatives exist and are equal, the function is differentiable at that point. Differentiability of Composite Functions
• To find the derivative of a composite function, we use the chain rule.
• The chain rule allows us to differentiate the outer function with respect to the inner function and multiply it by the derivative of the inner function. Example: Differentiability of Composite Function
• Let’s consider the function f(x) = (sin x)^2.
• The outer function is f(u) = u^2, and the inner function is u = sin x.
• To find the derivative of f(x), we differentiate the outer function with respect to the inner function and multiply it by the derivative of the inner function.
• By using the chain rule, we get f’(x) = 2(sin x)(cos x). Differentiability of Inverse Functions
• The derivative of an inverse function can be found using the formula: (f^(-1))’(x) = 1 / f’(f^(-1)(x))
• In other words, the derivative of the inverse function is the reciprocal of the derivative of the original function evaluated at the corresponding point.

  Example: Differentiability of Inverse Function

• Let’s consider the function f(x) = x^3 + 2x.
• The inverse function is f^(-1)(x).
• To find the derivative of the inverse function, we need to find the derivative of f(x) and evaluate it at f^(-1)(x).
• By differentiating f(x), we get f’(x) = 3x^2 + 2.
• The derivative of the inverse function is given by: (f^(-1))’(x) = 1 / (3(f^(-1)(x))^2 + 2). Relationship Between Differentiability and Continuity
• If a function is differentiable at a point, it must be continuous at that point.
• However, if a function is continuous at a point, it does not necessarily mean that it is differentiable at that point.
• We can have functions that are continuous but not differentiable, such as the absolute value function. Summary
• Differentiability at a point is determined by the existence of the derivative at that point.
• Differentiability implies continuity, but continuity does not guarantee differentiability.
• Most elementary functions are differentiable, and the sum, difference, product, and quotient of differentiable functions are also differentiable.
• The chain rule allows us to differentiate composite functions, and the inverse function rule helps us find the derivative of inverse functions.
• Piecewise functions may or may not be differentiable at the points where the pieces join.