derivates

1. Derivated number and tangent

a) Variation rate

$$\frac{f(b) - f(a)}{b-a}$$ $$\frac{\Delta{f}}{\Delta{x}}(a;b)$$

b) Derivated number of a function in a real $\mathbb{R}$

f is derivable only if: $$f'(a)=\lim_{h\to0}{\frac{f(a+h)-f(a)}{h}}$$

c) Tangent

A line passing through the point A, with a director coefficient of $f'(a)$ Equation: $$y=f'(a)(x-a)+f(a)$$

2. Derivated functions

a) On the $I$ interval

A function is derivable on $I$ if derivable for every $\mathbb{R}\in I$. Derivated function of $f(x)$ is noted $f'(x)$.

b) Reference functions

$f(x)=C$	$f'(x) = 0$
$f(x)=x$	$f'(x) = 1$
$f(x)=x^2$	$f'(x) = 2x$
$f(x)=\frac{1}{x}$	$f'(x) = -\frac{\normalsize1}{\normalsize{x^2}}$
$f(x)=\sqrt{x}$	$f'(x) = \frac{\normalsize1}{\normalsize2\sqrt{x}}$
$f(x)=x^n$	$f'(x) = nx^{n-1}$

Demonstration for $f(x) = x^2$

$$\frac{(a+h)^2 - a^2}{h} = \frac{a^2+2ah+h^2-a^2}{h} = \frac{2ah+h^2}{h} = 2a$$

3. Operations on derivation

a) Derivation of a sum of functions

If $u$ and $v$ are functions and are derivable on $I$, the sum of $(u+v)' = u'+v'$

b) Derivation of a multiplication of functions

If $u$ and $v$ are functions and are derivable on $I$, the product of $(u \times v)' = u'v + v'u$

b) Derivation of a quotient of functions

If $u$ and $v$ are functions and are derivable on $I$, the product of $(\frac{\normalsize u}{\normalsize v})' = \frac{\normalsize{u'v - v'u}}{\normalsize{v^2}}$

4. Composition of functions and derivation

$$f\in J, g \in I$$ $$h(x)=f(g(x))$$ $$x \longrightarrow g(x) = X \longrightarrow f(X)$$ $$h(x) = x \longrightarrow f(g(x))$$

Derivation of a composed function

$h'(x) = axf'a(ax+b)$ where $g(x) = ax+b$