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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]

1.Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability


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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

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