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.

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

f' (x) = u' (x) + v' (x) .

Likewise, a function multiplied by a factor

r

, i.e.

f : = r \cdot u

with

f (x) = (r \cdot u) (x) : = r \cdot u (x)

, is also differentiable, and we have

f' (x) = r \cdot u' (x) .

Using these two rules together with the differentiation rules for monomials

x^{n}

, any arbitrary polynomial can be differentiated. Here are some examples.

The polynomial

f

with the mapping rule

f (x) = \frac{1}{4} x^{3} - 2 x^{2} + 5

is differentiable, and we have

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

The derivative of the function

g :] 0; \infty [\to ℝ

with

g (x) = x^{3} + \ln (x)

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

Differentiating the function

h : [0; \infty [\to ℝ

with

h (x) = 4^{- 1} \cdot x^{2} - \sqrt{x} = \frac{1}{4} x^{2} + (- 1) \cdot x^{\frac{1}{2}}

results, for

x > 0

, in

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