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Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule
Let \(D\subseteq\mathbb R\) (\(D\) is a subset of real numbers). Let \(x\in D,\lambda\in\mathbb R\), and let \(f,g:D\to\mathbb R\) be differentiable functions at \(x\). Then
\[f+g,\quad fg,\quad \lambda f,\quad fg:D\to\mathbb R\]
are also differentiable at \(x\) and the following calculation rules apply:
 \((f+g)'(x)=f'(x)+g'(x)\),
 \((fg)'(x)=f'(x)g'(x)\),
 \((\lambda f)'(x)=\lambda f'(x)\),
 \((fg)'(x)=f'(x)g(x) + f(x)g'(x)\) (also known as the product rule).
 If \(g(\xi)\neq 0\) for all \(\xi\in D\), then also the function
\[\frac fg:D\to\mathbb R\]
is differentiable at \(x\) and the following formula (also known as the quotient rule) holds:
\[\left(\frac fg\right)'(x)=\frac{f'(x)g(x)  f(x)g'(x)}{g(x)^2}.\]
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
Propositions: 3
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983