Understanding Differentiability in Calculus | Definition, Properties, and Applications

If f is differentiable at x=a…

If a function f is differentiable at a point x = a, it means that the derivative of the function exists at that particular point

If a function f is differentiable at a point x = a, it means that the derivative of the function exists at that particular point. In other words, the function has a well-defined instantaneous rate of change at x = a.

The derivative of a function f(x) at a point x = a is denoted as f'(a) or dy/dx|a, where dy/dx represents the rate of change of the function with respect to x. The derivative essentially gives the slope of the tangent line to the graph of the function at the point of interest.

To determine whether a function is differentiable at a specific point, we need to check if the limit of the difference quotient exists as x approaches a. The difference quotient is defined as:

f'(a) = lim (h->0) [f(a + h) – f(a)] / h

If this limit exists and is finite, then the function is differentiable at x = a. It implies that the function has a well-defined instantaneous rate of change at that point.

Furthermore, differentiability at a point implies continuity at that point. If a function is differentiable at x = a, it must also be continuous at that point. However, continuity alone does not guarantee differentiability.

Differentiability is a fundamental concept in calculus and is used extensively to study the behavior of functions, finding maximum and minimum values, determining critical points, optimization, and more.

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