2024/G2/15

$\mathbf {3.3}$

$\lim _{\theta \to 0}{\frac {sin(\theta )}{\theta }}=1$
$\lim _{\theta \to 0}{\frac {cos(\theta )-1}{\theta }}=0$
${\frac {d}{dx}}[sin(x)]=cos(x)$

${\frac {d}{dx}}[cos(x)]=-sin(x)$

${\frac {d}{dx}}[tan(x)]=sec^{2}(x)$

${\frac {d}{dx}}[csc(x)]=-csc(x)\cdot cot(x)$

${\frac {d}{dx}}[sec(x)]=sec(x)\cdot tan(x)$

${\frac {d}{dx}}[cot(x)]=-csc^{2}(x)$

$\mathbf {\color {Blue}{Examples}}$

$\mathbf {\color {MidnightBlue}{Ex.2}}$
$f(x)={\frac {sec(x)}{1+tan(x)}}f^{\prime }(x)={\frac {[sec(x)tan(x)][1+tan(x)]-sec(x)[sec^{2}(x)]}{[1+tan(x)]^{2}}}={\frac {sec(x)[tan(x)+tan^{2}(x)-sec^{2}(x)]}{(1+tan(x))^{2}}}=sec(x)[tan(x)+tan^{2}(x)-[tan^{2}(x)+1]$