differentiierbare

Differentiierbare is a German adjective meaning "differentiable." In mathematics, differentiability is a fundamental concept related to the behavior of functions. A function is considered differentiable at a point if it has a derivative at that point. The derivative of a function at a point represents the instantaneous rate of change of the function at that point, or geometrically, the slope of the tangent line to the function's graph at that point.

For a function of a single real variable, $f(x)$, to be differentiable at a point $x_0$, the limit of the difference quotient must exist:

$$ \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} $$

If this limit exists, it is the derivative of $f$ at $x_0$, denoted as $f'(x_0)$.

In higher dimensions, for a function mapping from $\mathbb{R}^n$ to $\mathbb{R}^m$, differentiability is defined in terms of the existence of a linear transformation that best approximates the function's change near a point. This is closely related to the concept of partial derivatives and the Jacobian matrix.

Functions that are differentiable are also necessarily continuous, although the converse is not always true. For example, the absolute value function, $f(x) = |x|$, is continuous everywhere but not differentiable at $x=0$. Differentiability is a crucial property in many areas of calculus and analysis, including optimization, differential equations, and manifold theory, as it allows for the use of local linear approximations.