Chapter 7 Differential Calculus

Section 7.3 Calculation Rules

7.3.2 Multiples and Sums of Functions

In the following, u,v:D 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(x)=(u+v)(x):=u(x)+v(x) is also differentiable, and we have


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


Using these two rules together with the differentiation rules for monomials xn , any arbitrary polynomial can be differentiated. Here are some examples.
Example 7.3.2
The polynomial f with the mapping rule f(x)= 1 4 x3 -2 x2 +5 is differentiable, and we have

f'(x)= 3 4 x2 -4x.

The derivative of the function g: ]0;[ with g(x)= x3 +ln(x) is

g': ]0;[    with   g'(x)=3 x2 + 1 x = 3 x3 +1 x .

Differentiating the function h:[0;[ with h(x)= 4-1 · x2 -x= 1 4 x2 +(-1)· x 1 2 results, for x>0, in

h'(x)= 1 2 x- 1 2 x- 1 2 = x 3 2 -1 2x .