Selesaikan sinh(1)=

$$\sin h ( 1 ) =$$

Bezakan w.r.t. h

$$\cos(h)$$

Langkah Menggunakan Takrif Terbitan

Bagi fungsi f\left(x\right), terbitannya adalah had bagi \frac{f\left(x+h\right)-f\left(x\right)}{h} apabila h pergi ke 0, jika had tersebut wujud.

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

Gunakan Formula Hasil Tambah untuk Sinus.

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

Faktorkan \sin(h).

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

Tulis semula had.

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

Gunakan fakta bahawa h ialah pemalar apabila mengira had semasa t pergi ke 0.

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

Had \lim_{h\to 0}\frac{\sin(h)}{h} ialah 1.

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

Untuk menilaikan had \lim_{t\to 0}\frac{\cos(t)-1}{t}, mula-mula darabkan pengangka dan penyebut dengan \cos(t)+1.

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

Darabkan \cos(t)+1 kali \cos(t)-1.

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

Gunakan Identiti Phythagoras.

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

Tulis semula had.

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

Had \lim_{h\to 0}\frac{\sin(h)}{h} ialah 1.

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

Gunakan fakta bahawa \frac{\sin(t)}{\cos(t)+1} adalah selanjar pada 0.

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

Gantikan nilai 0 ke dalam ungkapan \sin(h)\left(\lim_{t\to 0}\frac{\cos(t)-1}{t}\right)+\cos(h).

$$\cos(h)$$

Nilaikan

$$\sin(h)$$