Differentiability of Real Functions Definition: A function $f$ is differentiable at a point $c$ if its derivative exists at that point. The derivative i...
Differentiability of Real Functions
Definition:
A function is differentiable at a point if its derivative exists at that point. The derivative is a limit of the difference quotient as the increment approaches 0.
Key Concepts:
Derivative: A measure of the instantaneous rate of change of a function at a given point.
Limit: A measure of the value a function approaches as its input approaches a specific value.
First Derivative: The derivative of a function is a function that gives the rate of change of the original function.
Higher-Order Derivatives: Derivatives of higher orders provide information about the function's behavior at higher rates.
Properties of Differentiability:
A function is differentiable at a point if it is continuous at that point.
The derivative of a constant function is always zero.
The derivative of a function of a function is equal to the function of the derivative of the original function.
Taylor's Theorem:
A function is differentiable at a point if and only if it is continuous at that point. Furthermore, the derivative of a differentiable function at a point is equal to the function itself.
Examples:
is differentiable at all points, with .
is differentiable at all points, but its derivative is not continuous at .
is not differentiable at , but it is differentiable at all other points.
Applications of Differentiability:
Finding critical points of functions
Determining intervals of increasing or decreasing values
Calculating the slope of a curve at a point
Approximating function values near a given point