Selesaikan cos(2x/2)

$$\cos ( \frac { 2 x } { 2 } )$$

Bezakan w.r.t. x

$$-\sin(x)$$

Langkah Menggunakan Takrif Terbitan

Batalkan 2 dan 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\cos(x))$$

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.

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

Gunakan Formula Hasil Tambah untuk Kosinus.

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

Faktorkan \cos(x).

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

Tulis semula had.

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

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

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

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

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

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

$$-\sin(x)$$

Nilaikan

$$\cos(x)$$