• Differentiability of a Function: A function f(x) is said to be differentiable at a point c if its derivative f(c) exists.

Conditions for Differentiability:

  1. Existence of Derivative: The limit limh0f(c+h)f(c)h exists.

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

Graphical Interpretation:

  • Tangent Line: A function is differentiable at a point if there exists a unique tangent line to the curve at that point.

  • Smooth Curve: Differentiability implies a smooth, non-cornered curve without abrupt changes.

Differentiability Rules:

  1. Sum/Difference Rule: (f±g)(x)=f(x)±g(x)

  2. Product Rule: (fg)(x)=f(x)g(x)+f(x)g(x)

  3. Quotient Rule: (fg)(x)=f(x)g(x)f(x)g(x)(g(x))2

  4. Chain Rule: (f(g(x)))=f(g(x))g(x)

Understanding Differentiability:

  • Smoothness of Function: Differentiability implies the function is smooth without abrupt changes.

  • Slope at a Point: Differentiability indicates the existence of a well-defined slope at a specific point on the curve.

Testing Differentiability:

  1. Using Derivative Definition: Calculating the derivative using the definition of the derivative.

  2. Applying Differentiability Rules: Employing derivative rules to find the derivative of a function.