Concept of Differentiability Definition: A function is differentiable at a point if its derivative exists at that point. The derivative is a measure...
Concept of Differentiability
Definition: A function is differentiable at a point if its derivative exists at that point. The derivative is a measure of how quickly the function's output changes with respect to changes in its input.
Key Concepts:
Derivative: A function's derivative is a special kind of limit called the derivative. The derivative is a number that represents how the function's output changes when its input changes by a small amount.
Slope: The slope is a measure of how quickly the function's output changes with respect to changes in its input. The slope is calculated by taking the limit of the difference quotient as the change in input approaches zero.
Differentiability Conditions: A function is differentiable at a point if its derivative exists at that point. If the derivative is continuous, the function is differentiable.
Examples:
A function like f(x) = x^2 is differentiable at any point.
A function like f(x) = 1/x is differentiable at x = 0 but not at x = 0.
A function like f(x) = sin(x) is differentiable for all real values of x.
Applications:
Differentiability is used in various applications, including:
Finding the rate of change of a function.
Determining the slope of a function at a given point.
Approximating the function's output for values close to a given point.
Additional Notes:
A function is continuous if its graph is smooth.
A function is differentiable if its graph has a defined slope at all points in its domain.
A function is derivable if it is differentiable and its derivative is continuous