2024/G1/3

3.2 THE PRODUCT AND QUOTIENT RULES

${\frac {d}{dx}}[c]=0$
${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]$

${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]$
${\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1$
${\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}$
${\frac {d}{dx}}[e^{x}]=e^{x}$

3.3 DERIVATIVE OF TRIGONOMETRIC FUNCTIONS

$\lim _{a\to 0}{\frac {sin(a)}{0}}=1$
$\lim _{a\to 0}{\frac {cos(a)}{0}}=0$
${\frac {d}{dx}}[sin(0)]=cos(0)$
${\frac {d}{dx}}[cos(0)]=-sin(0)$
${\frac {d}{dx}}[tan(0)]=sec^{2}(0)$
${\frac {d}{dx}}[csc(0)]=csc(0)\cdot cot(0)$
${\frac {d}{dx}}[sec(0)]=sec(0)\cdot tan(0)$
${\frac {d}{dx}}[cot(0)]=-csc^{2}(0)$

Ex.2

$f(x)={\frac {sec(x)}{1+tan(x)}}$
$F'(x)={\frac {[sec(x)\cdot tan(x)][1+tan(x)]-sec(x)[sec^{2}(x)}{[1+tan(x)]^{2}}}$
$F'(x)={\frac {sec(x)[tan(x)+tan^{2}(x)-sec^{2}(x)}{[1+tan(x)]^{2}}}$
$F'(x)={\frac {sec(x)(tan(x)+tan^{2}(x)-tan^{2}(x)-1)}{[1+tan(x)]^{2}}}$
$F'(x)={\frac {sec(x)[tan(x)-1]}{[1+tan(x)]^{2}}}$
$sec(x)=0\qquad tan(x)-1=0\qquad tan(x)=1$ caption

3.4

Point Slope Form

$y-y_{1}=m(x-x_{1})$