Let \(I\) be an open (not necessarily bounded) interval. We call a function \(f\colon I \to \mathbb{R}\) differentiable at \(x_0\in I\) if

\begin{equation*} f'(x_0)=\lim_{x \to x_0}\frac{f(x)-f(x_0)}{x-x_0} \end{equation*}

exists. This value is the derivative of \(f\) at \(x_0\).

If a function is differentiable at every point we simply call it differentiable and \(f'\) the derivative of \(f\).

Links Link to heading

• alternative ways of defining differentiability

Rules Link to heading

• Chain Rule

Results Link to heading

• mean value inequality
• differentiable Lipschitz functions have bounded derivative

Special Classes of Differentiable Functions Link to heading

• continuously differentiable functions \(C^k\)
• smooth functions \(C^\infty\)
• Sobolev functions \(W^{k,p}\)
• Schwartz functions \(\mathbb{S}\)
• analytic functions
• holomorphic functions

Relations to other concepts Link to heading

• continuity
• convolution
• Fourier transform

See also Link to heading

• multi-index notation for partial derivatives