3.3 {\displaystyle \mathbf {3.3} }
lim θ → 0 s i n ( θ ) θ = 1 {\displaystyle \lim _{\theta \to 0}{\frac {sin(\theta )}{\theta }}=1} lim θ → 0 c o s ( θ ) − 1 θ = 0 {\displaystyle \lim _{\theta \to 0}{\frac {cos(\theta )-1}{\theta }}=0} d d x [ s i n ( x ) ] = c o s ( x ) {\displaystyle {\frac {d}{dx}}[sin(x)]=cos(x)} d d x [ c o s ( x ) ] = − s i n ( x ) {\displaystyle {\frac {d}{dx}}[cos(x)]=-sin(x)} d d x [ t a n ( x ) ] = s e c 2 ( x ) {\displaystyle {\frac {d}{dx}}[tan(x)]=sec^{2}(x)} d d x [ c s c ( x ) ] = − c s c ( x ) ⋅ c o t ( x ) {\displaystyle {\frac {d}{dx}}[csc(x)]=-csc(x)\cdot cot(x)} d d x [ s e c ( x ) ] = s e c ( x ) ⋅ t a n ( x ) {\displaystyle {\frac {d}{dx}}[sec(x)]=sec(x)\cdot tan(x)} d d x [ c o t ( x ) ] = − c s c 2 ( x ) {\displaystyle {\frac {d}{dx}}[cot(x)]=-csc^{2}(x)}
E x a m p l e s {\displaystyle \mathbf {\color {Blue}{Examples}} }
E x .2 {\displaystyle \mathbf {\color {MidnightBlue}{Ex.2}} } f ( x ) = s e c ( x ) 1 + t a n ( x ) f ′ ( x ) = [ s e c ( x ) t a n ( x ) ] [ 1 + t a n ( x ) ] − s e c ( x ) [ s e c 2 ( x ) ] [ 1 + t a n ( x ) ] 2 = s e c ( x ) [ t a n ( x ) + t a n 2 ( x ) − s e c 2 ( x ) ] ( 1 + t a n ( x ) ) 2 = s e c ( x ) [ t a n ( x ) + t a n 2 ( x ) − [ t a n 2 ( x ) + 1 ] {\displaystyle 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]}
![{\displaystyle {\frac {d}{dx}}[sin(x)]=cos(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/63060492a2f660d277ff93c421d4619c309b1b0f)
![{\displaystyle {\frac {d}{dx}}[cos(x)]=-sin(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8a2f3c386f5555f5fb1c49aaa1ad0a3ac50931b8)
![{\displaystyle {\frac {d}{dx}}[tan(x)]=sec^{2}(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7fc70fbad633e6a4f9e695ac4037605f575c3f23)
![{\displaystyle {\frac {d}{dx}}[csc(x)]=-csc(x)\cdot cot(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7d8d8cb21ffc0489ed33c257b842bffb93751401)
![{\displaystyle {\frac {d}{dx}}[sec(x)]=sec(x)\cdot tan(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/beba7e3cd355bd25652be94ad15c785a56afc02d)
![{\displaystyle {\frac {d}{dx}}[cot(x)]=-csc^{2}(x)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/579de0a2a3c8934952b04011241a430532d93974)
![{\displaystyle 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]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5437559d00a241512b9d1c4991ef87623228db2c)