Issolvi sin(x2)

$$\sin ( x2 )$$

Iddifferenzja w.r.t. x_2

$$\cos(x_{2})$$

Passi għall-Użu ta' Definizzjoni ta' Dervattiv

Għall-funzjoni f\left(x\right), id-derivattiv huwa l-limitu ta' \frac{f\left(x+h\right)-f\left(x\right)}{h} billi h jmur għal 0, jekk jeżisti dak il-limitu.

$$\frac{\mathrm{d}}{\mathrm{d}x_{2}}(\sin(x_{2}))=\left(\lim_{h\to 0}\frac{\sin(x_{2}+h)-\sin(x_{2})}{h}\right)$$

Uża l-Formula tas-Somma għal Sine.

$$\lim_{h\to 0}\frac{\sin(x_{2}+h)-\sin(x_{2})}{h}$$

Iffattura 'l barra \sin(x_{2}).

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

Erġa' ikteb il-limitu.

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

Uża l-fatt li x_{2} huwa kostanti meta taħdem il-limiti bħala h jmorru għal 0.

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

Il-limitu \lim_{x_{2}\to 0}\frac{\sin(x_{2})}{x_{2}} huwa 1.

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

Biex tevalwa l-limitu \lim_{h\to 0}\frac{\cos(h)-1}{h}, l-ewwel immultiplika n-numeratur u d-denominatur b'\cos(h)+1.

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

Immultiplika \cos(h)+1 b'\cos(h)-1.

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

Uża l-Pythagorean Identity.

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

Erġa' ikteb il-limitu.

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

Il-limitu \lim_{x_{2}\to 0}\frac{\sin(x_{2})}{x_{2}} huwa 1.

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

Uża l-fatt li \frac{\sin(h)}{\cos(h)+1} huwa kontinwu f'0.

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

Issostitwixxi l-valur 0 fl-espressjoni \sin(x_{2})\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x_{2}).

$$\cos(x_{2})$$

Evalwa

$$\sin(x_{2})$$