Selesaikan d/dθleft(cos(θ)right)

$$\frac{d}{d \theta } \left( \cos ( \theta ) \right)$$

Nilaikan

$$-\sin(\theta )$$

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.

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

Gunakan Formula Hasil Tambah untuk Kosinus.

$$\lim_{h\to 0}\frac{\cos(h+\theta )-\cos(\theta )}{h}$$

Faktorkan \cos(\theta ).

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

Tulis semula had.

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

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

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

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

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

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_{\theta \to 0}\frac{\sin(\theta )}{\theta } 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 \cos(\theta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\sin(\theta ).

$$-\sin(\theta )$$

Bezakan w.r.t. θ

$$-\cos(\theta )$$