$$\sin w$$
$\cos(w)$
$$\frac{\mathrm{d}}{\mathrm{d}w}(\sin(w))=\left(\lim_{h\to 0}\frac{\sin(w+h)-\sin(w)}{h}\right)$$
$$\lim_{h\to 0}\frac{\sin(w+h)-\sin(w)}{h}$$
$$\lim_{h\to 0}\frac{\sin(w)\left(\cos(h)-1\right)+\cos(w)\sin(h)}{h}$$
$$\left(\lim_{h\to 0}\sin(w)\right)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\left(\lim_{h\to 0}\cos(w)\right)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$
$$\sin(w)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(w)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$
$$\sin(w)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(w)$$
$$\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)=\left(\lim_{h\to 0}\frac{\left(\cos(h)-1\right)\left(\cos(h)+1\right)}{h\left(\cos(h)+1\right)}\right)$$
$$\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}$$
$$\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}$$
$$\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$
$$-\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$
$$\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)=0$$
$$\cos(w)$$
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$\sin(w)$
