Derivatives

Definition 1. A function \(f\colon D\rightarrow W\) is called differentiable at \(x\in D\), if the limit \[\lim_{z\rightarrow x}\frac{f(z)-f(x)}{z-x}\] exists. The limit is called derivative at \(x\). One writes: \(f'(x)\), \(y'(x)\), \(\frac{\text{d}y}{\text{d}x}\), or \(\frac{\text{d}}{\text{d}x}f(x)\). A function \(f\colon D \rightarrow W\) is called differentiable, if it is differentiable at all points of the domain. The function \(f'\colon D\rightarrow W'\) is called derivative.