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Definition: Derivative, Differentiable Functions

Let $$D\subseteq\mathbb R$$ ($$D$$ is a subset of real numbers) and let $$f:D\to\mathbb R$$ be a function. Using the concept of the difference quotient,

$\Delta f(y,x)=\frac {f(y)-f(x)}{y-x},\quad y\neq x,$

for a fixed $$x\in D$$ and all $$\xi\in D$$ with $$\xi\neq x$$, we define a function $$g:D\to\mathbb R$$ by

$g(\xi):=\Delta f(\xi,x)=\frac {f(\xi)-f(x)}{\xi-x},\xi\neq x.$

If $$g$$ is continuous at $$x$$, its limit at $$x$$ exists1 and is unique2. This limit defines a new function

$f’(x):=\lim_{\substack{\xi\to x\\\xi\in D\setminus\{x\}}} g(\xi)=\lim_{\substack{\xi\to x\\\xi\in D\setminus\{x\}}}\frac {f(\xi)-f(x)}{\xi-x}.$

called the derivative of $$f$$ at $$x$$.

Other Notions of Derivatives

1. By a change of variables, the derivative $$f’(x)$$ can also be defined as $f’(x):=\lim_{h\to 0}\frac {f(x+h)-f(x)}{h}.$ In this definition, only those sequences $$(h_n)_{n\in\mathbb N}$$ with $$\lim h_n=0$$ are allowed, for which $$h_n\neq 0$$ and $$x+h_n\in D$$.
2. Instead of writing $$f’(x)$$, the derivative can also be written3 as $$\frac {d f(x)}{dx}$$.
3. $$f$$ has a derivative at $$x$$ $\Longleftrightarrow f$ is differentiable at $$x$$.
4. $$f$$ has a derivative at every $$x\in D$$ $$\Longleftrightarrow f$$ is differentiable on $D$.

1 Please note that the continuity of $$g$$ at $$x$$ requires the existence of at least one sequence $$(x_n)_{n\in\mathbb N}$$, $$x_n\in D$$ with $$\lim_{n\to\infty} x_n=x$$.

2 The uniqueness of the limit means that for every sequence $$(\xi_n)_{n\to\infty}$$, $$\xi_n\in D\setminus\{x\}$$ with $$\lim_{n\to\infty} \xi_n=x$$ we have $$g(\xi_n)=f’(x)$$.

3 The notation $$\frac {d f(x)}{dx}$$ can, in comparison with the notation $$f’(x)$$, sometimes become cumbersome. For instance, if $$x=0$$, we can write $$f’(0)$$, but we cannot write $$\frac {d f(0)}{d0}$$. In this case it is necessary to write $$\frac {d f}{dx}(0)$$ or $${\frac {d f(x)}{dx}}|^{d0}_{x=0}$$

| | | | | Contributors: bookofproofs | References: [581]

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