Proof

Let $$D\subseteq\mathbb R$$ ($$D$$ is a subset of real numbers). Let $$x\in D,\lambda\in\mathbb R$$. By hypothesis, $$f,g:D\to\mathbb R$$ are differentiable functions at $$x$$.

$$(1)$$ Proof of $$(f+g)'(x)=f'(x)+g'(x)$$

This rule follows immediately the definition of derivatives and the calculation rule for the sum of convergent real sequences.

$$(2)$$ Proof of $$(f-g)'(x)=f'(x)-g'(x)$$

This rule follows immediately the definition of derivatives and the calculation rule for the difference of convergent real sequences.

$$(3)$$ Proof of $$(\lambda f)'(x)=\lambda f'(x)$$

This rule follows immediately the definition of derivatives and the calculation rule for the product of a real number with a convergent real sequence.

$$(4)$$ Proof of the "Product Rule" $$(fg)'(x)=f'(x)g(x) + f(x)g'(x)$$

Applying the definition of derivatives we get

$\begin{array}{rcl} (fg)'(x)&=&\lim_{h\to 0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\ &=&\lim_{h\to 0}\frac{f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)}{h}\\ &=&\lim_{h\to 0}\frac{f(x+h)(g(x+h)-g(x))+(f(x+h)-f(x))g(x)}{h}\\ &=&\lim_{h\to 0}f(x+h)\frac{g(x+h)-g(x)}h+\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}g(x)\quad\quad( * )\\ &=&f'(x)g(x) + f(x)g'(x). \end{array}$

In the step $$( * )$$ we have used that $$f$$ is continuous, because it is differentiable at $$x$$.

$$(5)$$ Proof of the "Quotient Rule" $$\left(\frac fg\right)'(x)=\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$

By hypothesis, we have $$g(\xi)\neq 0$$ for all $$\xi\in D$$. We first consider the special case $$f=1$$.

$\begin{array}{rcl} \left(\frac 1g\right)'(x)&=&\lim_{h\to 0}\frac 1h\left(\frac 1{g(x+h)}- \frac 1{g(x)}\right)\\ &=&\lim_{h\to 0}\frac {1}{g(x+h)g(x)}\left(\frac {g(x)-g(x+h)}h\right)\\ &=&\frac {-g'(x)}{g(x)^2}. \end{array}$ Now, we can derive the general case by applying the Product Rule $$(4)$$ above:

$\begin{array}{rcl} \left(\frac fg\right)'(x)&=&\left(f\cdot \frac 1g\right)'(x)\\ &=&f'(x)\frac 1{g(x)} + f(x)\frac{-g'(x)}{g(x)^2}\\ &=&\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}. \end{array}$

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References

Bibliography

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