微分公式(6) 三角函數的微分
Author: 藍邦偉　老師

三角函數的微分

1

y=sinx

2

y=cosx

3

4

5

6

y=tanx

y=cotx

y=secx

y=cscx

f(x)=sinx

f'(x)=\lim_{ θ \to 0 }\frac{sin(x+θ)-sinx}{θ}

$ =\lim_{ θ \to 0 }\frac{(sinxcosθ+cosxsinθ)-sinx}{θ} $

$ =\lim_{ θ \to 0 }\left [  cosx\cdot \frac{sinθ}{θ}-sinx \left( \frac{1-cosθ}{θ}  \right) \right ]  $

$ =cosx\cdot \lim_{ θ \to 0 }\frac{sinθ}{θ}-sinx\cdot \lim_{ θ \to 0 }\frac{1-cosθ}{θ} $

$ =cosx\cdot 1-sinx\cdot 0 $

$ =cosx $

f(x)=cosx

f'(x)=\lim_{ θ \to 0 }\frac{cos(x+θ)-cosx}{θ}

$ =\lim_{ θ\to 0 }\frac{(cosxcosθ-sinxsinθ)-cosx}{θ} $

$ =\lim_{ θ \to 0 }\left [ cosx\cdot \left(\frac{cosθ-1}{θ}   \right)-sinx\cdot \left( \frac{sinθ}{θ}  \right)   \right ]  $

=cosx\cdot 0-sinx\cdot 1

=-sinx

f(x)=tanx=\left(  \frac{sinx}{cosx} \right) 

$ f'(x)=\frac{cosx\cdot cosx-(-sinx)\cdot sinx}{cos²x} $

$ =\frac{cos²x+sin²x}{cos²x} $

$= \frac{1}{cos²x} $

=sec²x

$ f(x)=cotx=\frac{cosx}{sinx} $

f'(x)=\frac{-sinx\cdot sinx-cosx\cdot cosx}{sin²x}

$ =\frac{-(sin²x+cos²x)}{sin²x} $

$ =\frac{-1}{sin²x} $

$ =-csc²x $

f(x)=secx=\frac{1}{cosx}

$ f'(x)=\frac{0\cdot cosx-(-sinx)\cdot 1}{cos²x} $

=\frac{sinx}{cosx}\cdot\frac{1}{cosx}

=tanx\cdot secx

f(x)=cscx=\frac{1}{sinx}

$ f'(x)=\frac{0\cdot sinx-cosx\cdot 1}{sin²x} $

=-\frac{cosx}{sinx}\cdot \frac{1}{sinx}

$ =-cotx\cdot cscx $

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