10.2 Derivatives of sums ‣ Chapter 10 Differential calculus ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 10 Differential calculus ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

Let’s suppose we have a function $f(x)=q(x)+r(x)$ , then the derivative of $f(x)$ is ⁶⁶ 6 Note that this relies on the property that the limit of two things added together is the same as the sum of the limits of the two things $\lim_{x\to a}(z(x)+q(x))=\lim_{x\to a}z(x)+\lim_{x\to a}q(x)$ Where $z(x)$ and $q(x)$ are any functions of $x$ whose limit is defined as $x\to a$ .

$\displaystyle\frac{df}{dx}$	$\displaystyle=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$	(10.11)
	$\displaystyle=\lim_{h\to 0}\frac{q(x+h)+r(x+h)-q(x)-r(x)}{h}$	(10.12)
	$\displaystyle=\lim_{h\to 0}\frac{q(x+h)-q(x)+r(x+h)-r(x)}{h}$	(10.13)
	$\displaystyle=\lim_{h\to 0}\frac{q(x+h)-q(x)}{h}+\lim_{h\to 0}\frac{r(x+h)-r(x% )}{h}$	(10.14)
	$\displaystyle=\frac{dq}{dx}+\frac{dr}{dx}$	(10.15)

That is to say that

\frac{d}{dx}(a(x)+b(x))=\frac{d}{dx}(a)+\frac{d}{dx}(b)

(10.16)