#### Chapter 7 Differential Calculus

Section 7.3 Calculation Rules

# 7.3.2 Multiples and Sums of Functions

In the following, $u,v:D\to ℝ$ will denote two arbitrary differentiable functions, and $r$ denotes an arbitrary real number.
##### Sum Rule and Constant Factor Rule 7.3.1
Let two differentiable functions $u$ and $v$ be given. Then, the sum $f:=u+v$ with $f\left(x\right)=\left(u+v\right)\left(x\right):=u\left(x\right)+v\left(x\right)$ is also differentiable, and we have

$f\text{'}\left(x\right)=u\text{'}\left(x\right)+v\text{'}\left(x\right) .$

Likewise, a function multiplied by a factor $r$, i.e. $f:=r·u$ with $f\left(x\right)=\left(r·u\right)\left(x\right):=r·u\left(x\right)$, is also differentiable, and we have

$f\text{'}\left(x\right)=r·u\text{'}\left(x\right) .$

Using these two rules together with the differentiation rules for monomials ${x}^{n}$, any arbitrary polynomial can be differentiated. Here are some examples.
##### Example 7.3.2
The polynomial $f$ with the mapping rule $f\left(x\right)=\frac{1}{4}{x}^{3}-2{x}^{2}+5$ is differentiable, and we have

$f\text{'}\left(x\right)=\frac{3}{4}{x}^{2}-4x .$

The derivative of the function $g:\text{}\right]0;\infty \left[\text{}\to ℝ$ with $g\left(x\right)={x}^{3}+\mathrm{ln}\left(x\right)$ is

$g\text{'}:\text{}\right]0;\infty \left[\text{}\to ℝ \mathrm{ }\text{with}\mathrm{ } g\text{'}\left(x\right)=3{x}^{2}+\frac{1}{x}=\frac{3{x}^{3}+1}{x} .$

Differentiating the function $h:\left[0;\infty \left[\text{}\to ℝ$ with $h\left(x\right)={4}^{-1}·{x}^{2}-\sqrt{x}=\frac{1}{4}{x}^{2}+\left(-1\right)·{x}^{\frac{1}{2}}$ results, for $x>0$, in

$h\text{'}\left(x\right)=\frac{1}{2}x-\frac{1}{2}{x}^{-\frac{1}{2}}=\frac{{x}^{\frac{3}{2}}-1}{2\sqrt{x}} .$