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

\[f+g,\quad f-g,\quad \lambda f,\quad fg:D\to\mathbb R\] are also differentiable at \(x\) and the following calculation rules apply:

  1. \((f+g)'(x)=f'(x)+g'(x)\),
  2. \((f-g)'(x)=f'(x)-g'(x)\),
  3. \((\lambda f)'(x)=\lambda f'(x)\),
  4. \((fg)'(x)=f'(x)g(x) + f(x)g'(x)\) (also known as the product rule).
  5. 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:

\[\left(\frac fg\right)'(x)=\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}.\]

Proofs: 1

Proofs: 1 2
Propositions: 3

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983