# 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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### References

#### Bibliography

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