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\frac{\mathrm{d}}{\mathrm{d}X}(\sin(X))=\left(\lim_{h\to 0}\frac{\sin(X+h)-\sin(X)}{h}\right)
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.
\lim_{h\to 0}\frac{\sin(X+h)-\sin(X)}{h}
Gunakan Formula Hasil Tambah untuk Sinus.
\lim_{h\to 0}\frac{\sin(X)\left(\cos(h)-1\right)+\cos(X)\sin(h)}{h}
Faktorkan \sin(X).
\left(\lim_{h\to 0}\sin(X)\right)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\left(\lim_{h\to 0}\cos(X)\right)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)
Tulis semula had.
\sin(X)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(X)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)
Gunakan fakta bahawa X ialah pemalar apabila mengira had semasa h pergi ke 0.
\sin(X)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(X)
Had \lim_{X\to 0}\frac{\sin(X)}{X} ialah 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)
Untuk menilaikan had \lim_{h\to 0}\frac{\cos(h)-1}{h}, mula-mula darabkan pengangka dan penyebut dengan \cos(h)+1.
\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}
Darabkan \cos(h)+1 kali \cos(h)-1.
\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}
Gunakan Identiti Phythagoras.
\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)
Tulis semula had.
-\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)=0
Gunakan fakta bahawa \frac{\sin(h)}{\cos(h)+1} adalah selanjar pada 0.
\cos(X)
Gantikan nilai 0 ke dalam ungkapan \sin(X)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(X).