Resolver sin(beta)

$$\sin ( \beta )$$

Diferentziatu β balioarekiko

$$\cos(\beta )$$

Deribatuaren definizioa erabiltzeko urratsak

f\left(x\right) funtzioetan, \frac{f\left(x+h\right)-f\left(x\right)}{h} funtzioaren limitea da deribatua, h 0 funtziora joaten baita, limitea baldin badago.

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

Erabili baturaren formula sinuan.

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

Deskonposatu \sin(\beta ).

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

Idatzi berriro limitea.

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

Erabili \beta konstantea limiteak kalkulatzean, h funtzioa 0 funtziora doan heinean.

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

\lim_{\beta \to 0}\frac{\sin(\beta )}{\beta } limitea 1 da.

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

\lim_{h\to 0}\frac{\cos(h)-1}{h} limitea ebaluatzeko, lehendabizi, biderkatu zenbakitzailea eta izendatzailea \cos(h)+1 balioarekin.

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

Egin \cos(h)+1 bider \cos(h)-1.

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

Erabili identitate pitagorikoa.

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

Idatzi berriro limitea.

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

\lim_{\beta \to 0}\frac{\sin(\beta )}{\beta } limitea 1 da.

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

Erabili \frac{\sin(h)}{\cos(h)+1} balioaren jarraitutasuna hemen: 0.

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

Ordezkatu 0 balioa \sin(\beta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\beta ) adierazpenarekin.

$$\cos(\beta )$$

Ebaluatu

$$\sin(\beta )$$