Whakaoti i te sinh(1)=

$$\sin h ( 1 ) =$$

Kimi Pārōnaki e ai ki h

$$\cos(h)$$

Ngā Upane e Whakamahi ana i te Tautuhinga o te Pārōnaki

Mō tētahi pānga f\left(x\right), ko te pārōnaki te tepe o \frac{f\left(x+h\right)-f\left(x\right)}{h} ina haere h ki 0, mēnā kei reira taua tepe.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\sin(h))=\left(\lim_{t\to 0}\frac{\sin(h+t)-\sin(h)}{t}\right)$$

Whakamahia te Tikanga Tātai Tapeke mō te Aho.

$$\lim_{t\to 0}\frac{\sin(t+h)-\sin(h)}{t}$$

Tauwehea te \sin(h).

$$\lim_{t\to 0}\frac{\sin(h)\left(\cos(t)-1\right)+\cos(h)\sin(t)}{t}$$

Tuhia anō te tepe.

$$\left(\lim_{t\to 0}\sin(h)\right)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\left(\lim_{t\to 0}\cos(h)\right)\left(\lim_{t\to 0}\frac{\sin(t)}{t}\right)$$

Whakamahia te meka ko h he pūmau ina tātai tepe i te wā ka haere t ki te 0.

$$\sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h)\left(\lim_{t\to 0}\frac{\sin(t)}{t}\right)$$

Ko te tepe \lim_{h\to 0}\frac{\sin(h)}{h} he 1.

$$\sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h)$$

Hei arotake i te tepe \lim_{t\to 0}\frac{\cos(t)-1}{t}, tuatahi me whakarea te taurunga me te tauraro ki te \cos(t)+1.

$$\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)=\left(\lim_{t\to 0}\frac{\left(\cos(t)-1\right)\left(\cos(t)+1\right)}{t\left(\cos(t)+1\right)}\right)$$

Whakareatia \cos(t)+1 ki te \cos(t)-1.

$$\lim_{t\to 0}\frac{\left(\cos(t)\right)^{2}-1}{t\left(\cos(t)+1\right)}$$

Whakamahia te Tuakiri Pythagorean.

$$\lim_{t\to 0}-\frac{\left(\sin(t)\right)^{2}}{t\left(\cos(t)+1\right)}$$

Tuhia anō te tepe.

$$\left(\lim_{t\to 0}-\frac{\sin(t)}{t}\right)\left(\lim_{t\to 0}\frac{\sin(t)}{\cos(t)+1}\right)$$

Ko te tepe \lim_{h\to 0}\frac{\sin(h)}{h} he 1.

$$-\left(\lim_{t\to 0}\frac{\sin(t)}{\cos(t)+1}\right)$$

Whakamahia te meka he motukore a \frac{\sin(t)}{\cos(t)+1} i 0.

$$\left(\lim_{t\to 0}\frac{\sin(t)}{\cos(t)+1}\right)=0$$

Whakakapihia te uara 0 ki roto i te kīanga \sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h).

$$\cos(h)$$

Aromātai

$$\sin(h)$$