PART 3：正弦、餘弦函數的微分(04:15)

定理3
\(f(x) = \sin x\) ，則 \({f^\prime }(x) = \cos x\)
證明: \({f^\prime }(x) = \lim\limits_{\Delta x \to 0} \frac{{\sin (x + \Delta x) - \sin x}}{{\Delta x}}\)
\(\lim\limits_{\Delta x \to 0} \frac{{\sin x\cos \Delta x + \cos x\sin \Delta x - \sin x}}{{\Delta x}}\)
\(\lim\limits_{\Delta x \to 0} \frac{{\sin x(\cos \Delta x - 1) + \cos x\sin \Delta x}}{{\Delta x}}\)
\(\lim\limits_{\Delta x \to 0} \frac{{\sin x(\cos \Delta x - 1)}}{{\Delta x}} + \lim\limits_{\Delta x \to 0} \cos x\frac{{\sin \Delta x}}{{\Delta x}}\)
\( = \cos x\)
(利用定理1與定理2，前面極限為0，後面極限為 \(\cos x\) )
定理4
\(f(x) = \cos x\) ，則 \({f^\prime }(x) =  - \sin x\)
(證明方法一樣，同學可自行練習)