Differentiability of a Function at a Point

Definition:

  • Differentiability at a Point c: A function f(x) is said to be differentiable at a point c in its domain if the derivative f(c) exists.

Conditions for Differentiability:

  1. Existence of the Derivative: The limit limh0f(c+h)f(c)h must exist for f(c) to exist.

  2. Continuity at the Point: If a function is differentiable at c, it must also be continuous at c.

Example:

Consider the function f(x)=x2.

Let's investigate the differentiability of f(x) at x=2.

Step 1: Calculating the Derivative:

The derivative of f(x)=x2 is f(x)=2x.

Step 2: Evaluating f(2):

Substituting x=2 into the derivative f(x)=2x:

f(2)=2×2=4

Step 3: Conclusion:

The derivative f(2)=4 exists, indicating that f(x)=x2 is differentiable at x=2.

Understanding Differentiability:

  • Derivative Existence: If the derivative of a function exists at a specific point, the function is differentiable at that point.

  • Tangent Line Interpretation: Differentiability implies the existence of a well-defined tangent line to the curve at that point.

Importance of Differentiability:

  • Rate of Change: Differentiability helps in understanding rates of change and slopes of functions.

  • Real-world Applications: Applied fields use differentiability to model various phenomena, such as motion, growth, and economics.