What is what does it mean if a function is differentiable?

A function is said to be differentiable at a point if its derivative exists at that point. Here's a breakdown of what that means:

• Existence of the Derivative: The core idea is that the <a href="https://www.wikiwhat.page/kavramlar/derivative%20of%20a%20function">derivative of a function</a>, f'(x), at a specific point x = a must exist. The derivative represents the instantaneous rate of change of the function at that point.

• The Limit Definition: The derivative is formally defined using a limit:

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

  For a function to be differentiable at x = a, this limit must exist and be finite. Crucially, the limit must be the same whether you approach a from the left or the right.

• Geometric Interpretation: Geometrically, differentiability implies that the function has a well-defined <a href="https://www.wikiwhat.page/kavramlar/tangent%20line">tangent line</a> at the point in question. You can draw a line that "just touches" the curve at that point, representing the direction the function is heading at that precise location.

• Continuity is a Prerequisite: Differentiability implies <a href="https://www.wikiwhat.page/kavramlar/continuity">continuity</a>. If a function is not continuous at a point, it cannot be differentiable at that point. However, the reverse is not true: a function can be continuous but not differentiable (e.g., a sharp corner).

• Points of Non-Differentiability: Functions are not differentiable at points where:

  • There is a sharp corner or cusp.
  • There is a vertical tangent line.
  • The function is discontinuous.

• Differentiability on an Interval: A function is differentiable on an open interval (a, b) if it is differentiable at every point within that interval. A function is differentiable on a closed interval [a, b] if it is differentiable on (a, b) and if the one-sided derivatives at the endpoints exist (i.e., the limit as h approaches 0 from the right at a and the limit as h approaches 0 from the left at b exist).