Selesaikan sin(beta)

$$\sin ( \beta )$$

Bezakan w.r.t. β

$$\cos(\beta )$$

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.

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

Gunakan Formula Hasil Tambah untuk Sinus.

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

Faktorkan \sin(\beta ).

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

Tulis semula had.

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

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

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

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

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

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

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

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

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

Gunakan Identiti Phythagoras.

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

Tulis semula had.

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

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

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

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

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

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

$$\cos(\beta )$$

Nilaikan

$$\sin(\beta )$$