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:
- 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.
- 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.
Continuity and Differentiability - Differentiability at a Point
Conditions for Differentiability at a Point
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The function must be defined at that point.
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The function must be continuous at that point.
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The derivative of the function must exist at that point.
Differentiability vs. Continuity -
Differentiability implies continuity, but continuity does not guarantee differentiability.
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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
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Let’s consider the function f(x) = x^2 + 3x - 2.
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The function is polynomial, and hence, it is differentiable for all values of x.
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To check the differentiability at a point, we need to find the derivative of f(x) and evaluate it at that point.
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By differentiating f(x), we get f’(x) = 2x + 3.
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Now, let’s find the derivative at x = 1.
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Substituting x = 1 in f’(x), we get f’(1) = 2(1) + 3 = 2 + 3 = 5.
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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.
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.