Analyse/Differentiatie: verschil tussen versies

& = \sin(x)\lim_{\Delta x \to 0}\frac{\Big(\cos(\Delta x)-1\Big)\Big(\cos(\Delta x)+1\Big)}{\Delta x \Big(\cos(\Delta x)+1\Big)} + \cos(x)\lim_{\Delta x \to 0}\frac{\sin(\Delta x)}{\Delta x} \\

& = \sin(x)\lim_{\Delta x \to 0}\frac{\Big(\cos^2(\Delta x)-1\Big)}{\Delta x\Big(\cos(\Delta x)+1\Big)} + \cos(x)\lim_{\Delta x \to 0}\frac{\sin(\Delta x)}{\Delta x} \\

& = \sin(x)\lim_{\Delta x \to 0}-\sin(\Delta x)\lim_{\Delta x \to 0}\frac{\sin(\Delta x)}{\Delta x}\lim_{\Delta x \to 0}\frac{1}{\cos(\Delta x)+1} + \cos(x)\lim_{\Delta x \to 0}\frac{\sin(\Delta x)}{\Delta x} \\

& = \sin(x) \cdot 0 \cdot 1 \cdot \frac{1}{2} + \cos(x) \cdot 1 \\