Atrisiniet sin(x2)

$$\sin ( x2 )$$

Diferencēt pēc x_2

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

Darbības, izmantojot atvasinājuma definīciju

Funkcijai f\left(x\right) atvasinājums ir \frac{f\left(x+h\right)-f\left(x\right)}{h} robeža, jo h sniedzas līdz 0, ja šāda robeža pastāv.

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

Izmantojiet sinusa summas formulu.

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

Iznesiet reizinātāju \sin(x_{2}) pirms iekavām.

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

Pārrakstiet robežu.

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

Aprēķinot robežas, izmantojiet faktu, ka x_{2} ir konstante, kad h tiecas uz 0.

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

Robeža \lim_{x_{2}\to 0}\frac{\sin(x_{2})}{x_{2}} ir 1.

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

Lai izrēķinātu robežu \lim_{h\to 0}\frac{\cos(h)-1}{h}, vispirms reiziniet skaitītāju un saucēju ar \cos(h)+1.

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

Reiziniet \cos(h)+1 reiz \cos(h)-1.

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

Izmantojiet Pitagora identitātes.

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

Pārrakstiet robežu.

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

Robeža \lim_{x_{2}\to 0}\frac{\sin(x_{2})}{x_{2}} ir 1.

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

Izmantojiet faktu, ka \frac{\sin(h)}{\cos(h)+1} ir nepārtraukta funkcija pie 0.

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

Aizstājiet vērtību 0 izteiksmē \sin(x_{2})\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x_{2}).

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

Izrēķināt

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